| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s + 13-s + 14-s + 16-s + 17-s − 6·19-s + 2·20-s + 7·23-s − 25-s + 26-s + 28-s − 29-s − 5·31-s + 32-s + 34-s + 2·35-s − 8·37-s − 6·38-s + 2·40-s − 10·41-s − 7·43-s + 7·46-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 + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.37·19-s + 0.447·20-s + 1.45·23-s − 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.185·29-s − 0.898·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s − 1.31·37-s − 0.973·38-s + 0.316·40-s − 1.56·41-s − 1.06·43-s + 1.03·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−14.65982327645694, −14.54196294993140, −13.66325932913143, −13.41916853599552, −12.98199909544681, −12.40980036708160, −11.87855185161089, −11.14222609047684, −10.93427079073780, −10.14832212688749, −9.897765130044225, −8.999726956680435, −8.604384584939815, −8.066225526404196, −7.181816263498654, −6.757772460625458, −6.295797289525090, −5.517766871087986, −5.178948365932125, −4.646749761954235, −3.751383737069597, −3.346467690833866, −2.442973468734043, −1.846642611119349, −1.337632871491669, 0,
1.337632871491669, 1.846642611119349, 2.442973468734043, 3.346467690833866, 3.751383737069597, 4.646749761954235, 5.178948365932125, 5.517766871087986, 6.295797289525090, 6.757772460625458, 7.181816263498654, 8.066225526404196, 8.604384584939815, 8.999726956680435, 9.897765130044225, 10.14832212688749, 10.93427079073780, 11.14222609047684, 11.87855185161089, 12.40980036708160, 12.98199909544681, 13.41916853599552, 13.66325932913143, 14.54196294993140, 14.65982327645694
