| L(s) = 1 | + 2-s − 4-s − 3·8-s + 2·13-s − 16-s − 6·17-s + 4·19-s − 4·23-s + 2·26-s + 6·29-s + 8·31-s + 5·32-s − 6·34-s − 6·37-s + 4·38-s − 2·41-s + 43-s − 4·46-s + 4·47-s − 7·49-s − 2·52-s − 2·53-s + 6·58-s + 14·61-s + 8·62-s + 7·64-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 0.392·26-s + 1.11·29-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 0.986·37-s + 0.648·38-s − 0.312·41-s + 0.152·43-s − 0.589·46-s + 0.583·47-s − 49-s − 0.277·52-s − 0.274·53-s + 0.787·58-s + 1.79·61-s + 1.01·62-s + 7/8·64-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.21088946320494694481179214932, −6.41402085488343175829415037737, −6.01196167735272552609084741177, −5.10334420175554881966055999306, −4.58147713901555743717388576335, −3.93962505623560800251964223941, −3.16622150941212496601878170079, −2.41388571808710428767146426114, −1.19707261635689080201634650011, 0,
1.19707261635689080201634650011, 2.41388571808710428767146426114, 3.16622150941212496601878170079, 3.93962505623560800251964223941, 4.58147713901555743717388576335, 5.10334420175554881966055999306, 6.01196167735272552609084741177, 6.41402085488343175829415037737, 7.21088946320494694481179214932
