| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s + 4·11-s + 14-s + 16-s − 17-s − 4·19-s − 2·20-s − 4·22-s + 4·23-s − 25-s − 28-s − 6·29-s − 32-s + 34-s + 2·35-s + 2·37-s + 4·38-s + 2·40-s − 2·41-s − 4·43-s + 4·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s + 0.834·23-s − 1/5·25-s − 0.188·28-s − 1.11·29-s − 0.176·32-s + 0.171·34-s + 0.338·35-s + 0.328·37-s + 0.648·38-s + 0.316·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8129024985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8129024985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.41053909996768, −11.99659968115016, −11.39585223507117, −11.23270593674012, −10.90797581031490, −10.11686014439820, −9.658330258386992, −9.485718677006805, −8.726709610683577, −8.453297178671885, −8.143720370848873, −7.310675092652178, −7.139572020330280, −6.596838150399957, −6.227047794057421, −5.623284823823262, −4.958636779540969, −4.388032308663865, −3.819896455950880, −3.534172828888181, −2.932470013997604, −2.137096790852301, −1.706595116160242, −0.9488989964757502, −0.3080717990368132,
0.3080717990368132, 0.9488989964757502, 1.706595116160242, 2.137096790852301, 2.932470013997604, 3.534172828888181, 3.819896455950880, 4.388032308663865, 4.958636779540969, 5.623284823823262, 6.227047794057421, 6.596838150399957, 7.139572020330280, 7.310675092652178, 8.143720370848873, 8.453297178671885, 8.726709610683577, 9.485718677006805, 9.658330258386992, 10.11686014439820, 10.90797581031490, 11.23270593674012, 11.39585223507117, 11.99659968115016, 12.41053909996768
