| L(s) = 1 | − 8·2-s + 34·4-s + 10·5-s − 14·7-s − 96·8-s − 80·10-s + 14·11-s + 50·13-s + 112·14-s + 196·16-s + 50·17-s + 36·19-s + 340·20-s − 112·22-s − 244·23-s + 75·25-s − 400·26-s − 476·28-s + 26·29-s − 120·31-s − 352·32-s − 400·34-s − 140·35-s + 564·37-s − 288·38-s − 960·40-s + 328·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 17/4·4-s + 0.894·5-s − 0.755·7-s − 4.24·8-s − 2.52·10-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 + 3.80·20-s − 1.08·22-s − 2.21·23-s + 3/5·25-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 − 0.676·35-s + 2.50·37-s − 1.22·38-s − 3.79·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7952317578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7952317578\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 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
−11.05045810834759882923289655509, −10.82595620169439374386044625293, −10.03506708339436406126858493542, −9.991102482284658848926355405589, −9.589197403687613012345096762524, −9.256755215362388152699174337459, −8.765080844569788964574115615407, −8.250511204401108437830195045069, −7.85796112771084375656852881012, −7.51468412112048442240412634178, −6.67226643040837680714918820400, −6.39462802812522710275969801258, −5.85166306504775962032271663219, −5.33990507519871392978892684932, −4.00330066673777330629276434001, −3.55385585016490645879943977057, −2.35611731956218467381565749202, −2.02065331865541195418603221681, −0.878670259546278060408011684066, −0.75815921534852244250859828078,
0.75815921534852244250859828078, 0.878670259546278060408011684066, 2.02065331865541195418603221681, 2.35611731956218467381565749202, 3.55385585016490645879943977057, 4.00330066673777330629276434001, 5.33990507519871392978892684932, 5.85166306504775962032271663219, 6.39462802812522710275969801258, 6.67226643040837680714918820400, 7.51468412112048442240412634178, 7.85796112771084375656852881012, 8.250511204401108437830195045069, 8.765080844569788964574115615407, 9.256755215362388152699174337459, 9.589197403687613012345096762524, 9.991102482284658848926355405589, 10.03506708339436406126858493542, 10.82595620169439374386044625293, 11.05045810834759882923289655509
