| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 14-s − 15-s + 16-s − 3·17-s + 18-s − 8·19-s − 20-s + 21-s − 2·22-s − 4·23-s + 24-s − 4·25-s + 27-s + 28-s + 9·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 1.67·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 1063 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−7.893004439582035985618602476878, −6.67760924838015052475931440534, −6.52474543291886715336481473209, −5.37485490918330349945978591486, −4.56663905163314047370053708147, −4.16480447023665740646878254286, −3.24244410511387649103496938963, −2.40150923742754578445903078572, −1.70546663526220721501484467777, 0,
1.70546663526220721501484467777, 2.40150923742754578445903078572, 3.24244410511387649103496938963, 4.16480447023665740646878254286, 4.56663905163314047370053708147, 5.37485490918330349945978591486, 6.52474543291886715336481473209, 6.67760924838015052475931440534, 7.893004439582035985618602476878
