| L(s) = 1 | − 6·2-s + 2·3-s + 13·4-s − 10·5-s − 12·6-s + 12·8-s − 33·9-s + 60·10-s + 28·11-s + 26·12-s + 36·13-s − 20·15-s − 147·16-s + 76·17-s + 198·18-s − 160·19-s − 130·20-s − 168·22-s − 22·23-s + 24·24-s + 75·25-s − 216·26-s − 86·27-s − 250·29-s + 120·30-s + 132·31-s + 366·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.384·3-s + 13/8·4-s − 0.894·5-s − 0.816·6-s + 0.530·8-s − 1.22·9-s + 1.89·10-s + 0.767·11-s + 0.625·12-s + 0.768·13-s − 0.344·15-s − 2.29·16-s + 1.08·17-s + 2.59·18-s − 1.93·19-s − 1.45·20-s − 1.62·22-s − 0.199·23-s + 0.204·24-s + 3/5·25-s − 1.62·26-s − 0.612·27-s − 1.60·29-s + 0.730·30-s + 0.764·31-s + 2.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p T + 23 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 37 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 2658 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 36 T + 4518 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 76 T + 5438 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 160 T + 18766 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 22 T - 2923 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 250 T + 57203 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 132 T + 43938 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 416 T + 107578 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 106 T + 138851 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 666 T + 269853 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 196 T + 209058 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 952 T + 484002 T^{2} + 952 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 840 T + 445646 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 98 T - 193437 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1286 T + 1005453 T^{2} + 1286 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1064 T + 753846 T^{2} - 1064 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 172 T + 757582 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1240 T + 1278886 T^{2} + 1240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1906 T + 2051733 T^{2} - 1906 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 650 T + 1305611 T^{2} + 650 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 628 T + 1423942 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.00447198868702718049414150212, −10.97270330664543862439346523607, −10.28061049917572038710589505732, −9.906659673113176271002140137805, −9.105818861746658644175725822508, −9.017159795864714526674431420318, −8.405125639031050811305536291883, −8.367496780956048721166456507188, −7.65400117964934738215036689460, −7.43586182918323600288200595072, −6.37438052289435044536236668632, −6.22068040978213335311116041835, −5.12052218472154734965649134747, −4.47875616141197994451195542212, −3.63623616393857740721138000635, −3.24898798493019027963937352291, −1.93756783274652489068805309255, −1.31414350096008295868992079198, 0, 0,
1.31414350096008295868992079198, 1.93756783274652489068805309255, 3.24898798493019027963937352291, 3.63623616393857740721138000635, 4.47875616141197994451195542212, 5.12052218472154734965649134747, 6.22068040978213335311116041835, 6.37438052289435044536236668632, 7.43586182918323600288200595072, 7.65400117964934738215036689460, 8.367496780956048721166456507188, 8.405125639031050811305536291883, 9.017159795864714526674431420318, 9.105818861746658644175725822508, 9.906659673113176271002140137805, 10.28061049917572038710589505732, 10.97270330664543862439346523607, 11.00447198868702718049414150212
