| L(s) = 1 | + 270·5-s − 1.11e3·7-s − 5.72e3·11-s + 4.57e3·13-s + 7.31e4·17-s + 1.03e5·19-s + 2.22e4·23-s + 7.81e4·25-s − 1.57e5·29-s + 1.03e5·31-s − 3.00e5·35-s − 1.89e5·37-s + 6.59e5·41-s + 7.57e4·43-s + 4.05e5·47-s + 8.23e5·49-s + 2.69e6·53-s − 1.54e6·55-s − 1.30e6·59-s − 1.83e6·61-s + 1.23e6·65-s − 1.36e6·67-s − 5.42e6·71-s + 5.73e6·73-s + 6.36e6·77-s + 1.12e6·79-s + 5.91e6·83-s + ⋯ |
| L(s) = 1 | + 0.965·5-s − 1.22·7-s − 1.29·11-s + 0.576·13-s + 3.60·17-s + 3.46·19-s + 0.381·23-s + 25-s − 1.19·29-s + 0.626·31-s − 1.18·35-s − 0.615·37-s + 1.49·41-s + 0.145·43-s + 0.569·47-s + 49-s + 2.48·53-s − 1.25·55-s − 0.826·59-s − 1.03·61-s + 0.557·65-s − 0.556·67-s − 1.79·71-s + 1.72·73-s + 1.58·77-s + 0.257·79-s + 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.439834281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.439834281\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 54 p T - 209 p^{2} T^{2} - 54 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 1112 T + 413001 T^{2} + 1112 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5724 T + 13277005 T^{2} + 5724 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4570 T - 41863617 T^{2} - 4570 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 36558 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 51740 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22248 T - 2909851943 T^{2} - 22248 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 157194 T + 7460077327 T^{2} + 157194 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 103936 T - 16709922015 T^{2} - 103936 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 94834 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 659610 T + 240331078219 T^{2} - 659610 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 75772 T - 266077215123 T^{2} - 75772 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 405648 T - 342072820559 T^{2} - 405648 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 1346274 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1303884 T - 788537999363 T^{2} + 1303884 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 30062 p T + 59127543 p^{2} T^{2} + 30062 p^{8} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1369388 T - 4185488110779 T^{2} + 1369388 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2714040 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2868794 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1129648 T - 17927804382255 T^{2} - 1129648 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5912028 T + 7816024083157 T^{2} - 5912028 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 897750 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 13719074 T + 107414706939363 T^{2} + 13719074 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56392866974299605191210860764, −10.00273250955862679710543480727, −9.803591321411487307417050105740, −9.258441615689266678010127558193, −9.188316154563624660510064156335, −8.043042565145177621027376880937, −7.75354053390204660698630993177, −7.36227677044067718001478223913, −6.96112603095881301256105021083, −5.84550600260590095084995491749, −5.82821295422280275736118769452, −5.35798134142339307123894562989, −5.12030020839351020817776856784, −3.82391360630994905915432423420, −3.35013819818144377625950323624, −2.95533696904943995896361258957, −2.63741820065074179386110783480, −1.37864573773758544801381966116, −1.07098244400134914634534691907, −0.60119573003290960376206579767,
0.60119573003290960376206579767, 1.07098244400134914634534691907, 1.37864573773758544801381966116, 2.63741820065074179386110783480, 2.95533696904943995896361258957, 3.35013819818144377625950323624, 3.82391360630994905915432423420, 5.12030020839351020817776856784, 5.35798134142339307123894562989, 5.82821295422280275736118769452, 5.84550600260590095084995491749, 6.96112603095881301256105021083, 7.36227677044067718001478223913, 7.75354053390204660698630993177, 8.043042565145177621027376880937, 9.188316154563624660510064156335, 9.258441615689266678010127558193, 9.803591321411487307417050105740, 10.00273250955862679710543480727, 10.56392866974299605191210860764
