| L(s) = 1 | + 8·2-s + 34·4-s + 14·7-s + 96·8-s + 14·11-s − 50·13-s + 112·14-s + 196·16-s − 50·17-s + 36·19-s + 112·22-s + 244·23-s − 400·26-s + 476·28-s + 26·29-s − 120·31-s + 352·32-s − 400·34-s − 564·37-s + 288·38-s + 328·41-s + 260·43-s + 476·44-s + 1.95e3·46-s − 350·47-s + 147·49-s − 1.70e3·52-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 17/4·4-s + 0.755·7-s + 4.24·8-s + 0.383·11-s − 1.06·13-s + 2.13·14-s + 3.06·16-s − 0.713·17-s + 0.434·19-s + 1.08·22-s + 2.21·23-s − 3.01·26-s + 3.21·28-s + 0.166·29-s − 0.695·31-s + 1.94·32-s − 2.01·34-s − 2.50·37-s + 1.22·38-s + 1.24·41-s + 0.922·43-s + 1.63·44-s + 6.25·46-s − 1.08·47-s + 3/7·49-s − 4.53·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(21.43668216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.43668216\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{3} T + 15 p T^{2} - p^{6} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 14 T + 663 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 50 T + 4987 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 50 T + 387 p T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T + 10170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 244 T + 29970 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 26 T + 47795 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 120 T - 1618 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 564 T + 173630 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 p T + 133986 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 260 T + 166666 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 350 T + 203423 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 56 T + 265770 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 616 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 336 T + 458858 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 152 T + 599110 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 952 T + p^{3} T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 676 T + 655606 T^{2} + 676 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 216 T + 1417730 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2742 T + 3608187 T^{2} + 2742 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.137782411391100420742336774239, −8.881982939758830979889589943622, −8.437540764171217802569233428049, −7.84870358613648712482921814105, −7.29298981146451621725259548094, −7.09503321598802074787734856127, −6.53512913539523144555875267523, −6.51244894676435169638107792322, −5.49208933703600007064020239737, −5.43981683249777740638572004133, −5.04026255138158721014435521234, −4.86421566479748640088953277247, −4.25417421365885802669837411108, −3.89834350273012229687821583878, −3.44528785216566537713760772089, −3.03848523496001632872775020048, −2.30809900767388224883449471779, −2.21941165757773480385632616640, −1.28656423209930191114592692801, −0.60335729966748845531091913640,
0.60335729966748845531091913640, 1.28656423209930191114592692801, 2.21941165757773480385632616640, 2.30809900767388224883449471779, 3.03848523496001632872775020048, 3.44528785216566537713760772089, 3.89834350273012229687821583878, 4.25417421365885802669837411108, 4.86421566479748640088953277247, 5.04026255138158721014435521234, 5.43981683249777740638572004133, 5.49208933703600007064020239737, 6.51244894676435169638107792322, 6.53512913539523144555875267523, 7.09503321598802074787734856127, 7.29298981146451621725259548094, 7.84870358613648712482921814105, 8.437540764171217802569233428049, 8.881982939758830979889589943622, 9.137782411391100420742336774239
