L(s) = 1 | + 4·7-s − 6·9-s − 4·13-s − 6·17-s − 5·23-s + 3·27-s + 12·29-s + 18·31-s − 10·37-s − 6·41-s + 10·43-s + 22·47-s − 2·49-s − 10·53-s + 59-s + 10·61-s − 24·63-s + 8·67-s − 8·71-s − 6·73-s + 9·81-s − 2·83-s + 14·89-s − 16·91-s − 6·97-s − 3·101-s + 2·103-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 2·9-s − 1.10·13-s − 1.45·17-s − 1.04·23-s + 0.577·27-s + 2.22·29-s + 3.23·31-s − 1.64·37-s − 0.937·41-s + 1.52·43-s + 3.20·47-s − 2/7·49-s − 1.37·53-s + 0.130·59-s + 1.28·61-s − 3.02·63-s + 0.977·67-s − 0.949·71-s − 0.702·73-s + 81-s − 0.219·83-s + 1.48·89-s − 1.67·91-s − 0.609·97-s − 0.298·101-s + 0.197·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.863934802\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.863934802\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{5} \) |
good | 3 | $C_2 \wr S_5$ | \( 1 + 2 p T^{2} - p T^{3} + p^{3} T^{4} - 2 p T^{5} + p^{4} T^{6} - p^{3} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 4 T + 18 T^{2} - 36 T^{3} + 153 T^{4} - 312 T^{5} + 153 p T^{6} - 36 p^{2} T^{7} + 18 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 40 T^{2} + 6 T^{3} + 763 T^{4} + 108 T^{5} + 763 p T^{6} + 6 p^{2} T^{7} + 40 p^{3} T^{8} + p^{5} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 4 T + 33 T^{2} + 9 p T^{3} + 729 T^{4} + 2037 T^{5} + 729 p T^{6} + 9 p^{3} T^{7} + 33 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 6 T + 37 T^{2} + 48 T^{3} - 230 T^{4} - 2220 T^{5} - 230 p T^{6} + 48 p^{2} T^{7} + 37 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 80 T^{2} + 6 T^{3} + 2803 T^{4} + 204 T^{5} + 2803 p T^{6} + 6 p^{2} T^{7} + 80 p^{3} T^{8} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 12 T + 5 p T^{2} - 1053 T^{3} + 8137 T^{4} - 43677 T^{5} + 8137 p T^{6} - 1053 p^{2} T^{7} + 5 p^{4} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 18 T + 8 p T^{2} - 2277 T^{3} + 17743 T^{4} - 106350 T^{5} + 17743 p T^{6} - 2277 p^{2} T^{7} + 8 p^{4} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 10 T + 153 T^{2} + 984 T^{3} + 9210 T^{4} + 46044 T^{5} + 9210 p T^{6} + 984 p^{2} T^{7} + 153 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 6 T + 139 T^{2} + 747 T^{3} + 10003 T^{4} + 41535 T^{5} + 10003 p T^{6} + 747 p^{2} T^{7} + 139 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 10 T + 192 T^{2} - 1368 T^{3} + 15411 T^{4} - 82140 T^{5} + 15411 p T^{6} - 1368 p^{2} T^{7} + 192 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 22 T + 332 T^{2} - 3835 T^{3} + 35433 T^{4} - 263290 T^{5} + 35433 p T^{6} - 3835 p^{2} T^{7} + 332 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 10 T + 173 T^{2} + 1720 T^{3} + 14166 T^{4} + 125884 T^{5} + 14166 p T^{6} + 1720 p^{2} T^{7} + 173 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - T + 212 T^{2} - 343 T^{3} + 21441 T^{4} - 30508 T^{5} + 21441 p T^{6} - 343 p^{2} T^{7} + 212 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 10 T + 105 T^{2} - 1368 T^{3} + 13410 T^{4} - 84828 T^{5} + 13410 p T^{6} - 1368 p^{2} T^{7} + 105 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 8 T + 147 T^{2} + 264 T^{3} - 1602 T^{4} + 116928 T^{5} - 1602 p T^{6} + 264 p^{2} T^{7} + 147 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 8 T + 176 T^{2} + 1583 T^{3} + 18243 T^{4} + 136046 T^{5} + 18243 p T^{6} + 1583 p^{2} T^{7} + 176 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 6 T + 143 T^{2} + 1713 T^{3} + 15079 T^{4} + 157581 T^{5} + 15079 p T^{6} + 1713 p^{2} T^{7} + 143 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 242 T^{2} + 36 T^{3} + 31333 T^{4} + 7416 T^{5} + 31333 p T^{6} + 36 p^{2} T^{7} + 242 p^{3} T^{8} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 2 T + 224 T^{2} - 448 T^{3} + 21855 T^{4} - 99532 T^{5} + 21855 p T^{6} - 448 p^{2} T^{7} + 224 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 14 T + 401 T^{2} - 3704 T^{3} + 62598 T^{4} - 434420 T^{5} + 62598 p T^{6} - 3704 p^{2} T^{7} + 401 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 6 T + 341 T^{2} + 1512 T^{3} + 52762 T^{4} + 180324 T^{5} + 52762 p T^{6} + 1512 p^{2} T^{7} + 341 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.47479220252862822725090599571, −4.47042748819660024342300079008, −4.39868135519520081750600442904, −4.24276242643006600286926932147, −4.08822879813071729181069765799, −3.55877249094041979577900042782, −3.52367352477365884582138162156, −3.46733897305509890298865536426, −3.30528955055190346275381407283, −3.19400314808820152415582066922, −2.71031038421101449106932691959, −2.68010365462465809378320799834, −2.61744796507593081688502258421, −2.56083861479817338943796745808, −2.38537825222669402793034432963, −2.18980538243705748775337690397, −1.94059712860206779292484478034, −1.80263076490724008514860425897, −1.54483661665748764587662467174, −1.42549158388150270482668883960, −1.04145635972342724304465882447, −0.891467963584972448314230439897, −0.49159643727688352371154610792, −0.49022438132558114992486558472, −0.37984942254883052554188979365,
0.37984942254883052554188979365, 0.49022438132558114992486558472, 0.49159643727688352371154610792, 0.891467963584972448314230439897, 1.04145635972342724304465882447, 1.42549158388150270482668883960, 1.54483661665748764587662467174, 1.80263076490724008514860425897, 1.94059712860206779292484478034, 2.18980538243705748775337690397, 2.38537825222669402793034432963, 2.56083861479817338943796745808, 2.61744796507593081688502258421, 2.68010365462465809378320799834, 2.71031038421101449106932691959, 3.19400314808820152415582066922, 3.30528955055190346275381407283, 3.46733897305509890298865536426, 3.52367352477365884582138162156, 3.55877249094041979577900042782, 4.08822879813071729181069765799, 4.24276242643006600286926932147, 4.39868135519520081750600442904, 4.47042748819660024342300079008, 4.47479220252862822725090599571