| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 11-s + 12-s + 2·13-s − 2·14-s + 16-s − 2·17-s + 18-s − 4·19-s − 2·21-s − 22-s + 8·23-s + 24-s − 5·25-s + 2·26-s + 27-s − 2·28-s − 10·29-s + 2·31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.436·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s − 0.377·28-s − 1.85·29-s + 0.359·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.16295275413321, −13.36549462973676, −13.20656991721765, −12.83502153142455, −12.43310863176048, −11.55360666117027, −11.25342659973626, −10.66401806628389, −10.27276348935197, −9.529752500551527, −9.063118143561899, −8.765777719787874, −7.943266321045821, −7.451239214547850, −7.045953629274904, −6.309281339094258, −6.014259456232017, −5.338957073680210, −4.715055653338689, −3.964062798190424, −3.757566274087080, −2.991017190354388, −2.472939158734554, −1.900772045332961, −1.015832983531596, 0,
1.015832983531596, 1.900772045332961, 2.472939158734554, 2.991017190354388, 3.757566274087080, 3.964062798190424, 4.715055653338689, 5.338957073680210, 6.014259456232017, 6.309281339094258, 7.045953629274904, 7.451239214547850, 7.943266321045821, 8.765777719787874, 9.063118143561899, 9.529752500551527, 10.27276348935197, 10.66401806628389, 11.25342659973626, 11.55360666117027, 12.43310863176048, 12.83502153142455, 13.20656991721765, 13.36549462973676, 14.16295275413321
