| L(s) = 1 | − 8·2-s − 2·3-s + 34·4-s + 16·6-s + 14·7-s − 96·8-s − 19·9-s − 14·11-s − 68·12-s − 50·13-s − 112·14-s + 196·16-s + 50·17-s + 152·18-s + 36·19-s − 28·21-s + 112·22-s − 244·23-s + 192·24-s + 400·26-s + 30·27-s + 476·28-s − 26·29-s − 120·31-s − 352·32-s + 28·33-s − 400·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.384·3-s + 17/4·4-s + 1.08·6-s + 0.755·7-s − 4.24·8-s − 0.703·9-s − 0.383·11-s − 1.63·12-s − 1.06·13-s − 2.13·14-s + 3.06·16-s + 0.713·17-s + 1.99·18-s + 0.434·19-s − 0.290·21-s + 1.08·22-s − 2.21·23-s + 1.63·24-s + 3.01·26-s + 0.213·27-s + 3.21·28-s − 0.166·29-s − 0.695·31-s − 1.94·32-s + 0.147·33-s − 2.01·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p^{3} T + 15 p T^{2} + p^{6} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−11.80825087439360764223821585532, −11.28505152687293393368280636519, −10.53332653781250101704097295266, −10.49534608207034099304100668135, −9.967465551565138258653644558968, −9.495296333070899946858132608086, −8.742574564517385533197989912519, −8.733178217326145290307301973842, −7.82242308001519031268868003098, −7.70976338683753046062229655833, −7.27516159588567079769821899657, −6.46369104127835772218170227884, −5.55668888252589109026295256726, −5.32122850061756159559472380459, −4.13771555854755557282415385814, −2.99085211354196282624221812754, −1.99036491990729153464656318823, −1.43820304951724344161710818402, 0, 0,
1.43820304951724344161710818402, 1.99036491990729153464656318823, 2.99085211354196282624221812754, 4.13771555854755557282415385814, 5.32122850061756159559472380459, 5.55668888252589109026295256726, 6.46369104127835772218170227884, 7.27516159588567079769821899657, 7.70976338683753046062229655833, 7.82242308001519031268868003098, 8.733178217326145290307301973842, 8.742574564517385533197989912519, 9.495296333070899946858132608086, 9.967465551565138258653644558968, 10.49534608207034099304100668135, 10.53332653781250101704097295266, 11.28505152687293393368280636519, 11.80825087439360764223821585532
