| L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s − 5·11-s − 6·13-s + 2·14-s + 16-s + 4·17-s + 20-s + 5·22-s − 23-s − 4·25-s + 6·26-s − 2·28-s − 9·29-s + 8·31-s − 32-s − 4·34-s − 2·35-s − 8·37-s − 40-s − 2·41-s − 5·43-s − 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.50·11-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.223·20-s + 1.06·22-s − 0.208·23-s − 4/5·25-s + 1.17·26-s − 0.377·28-s − 1.67·29-s + 1.43·31-s − 0.176·32-s − 0.685·34-s − 0.338·35-s − 1.31·37-s − 0.158·40-s − 0.312·41-s − 0.762·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.56770205675289, −13.09109149383375, −12.53781220120570, −12.10182805350026, −11.85223355255401, −11.01575041549171, −10.49221597204801, −10.12680594221381, −9.732162655585116, −9.529589595090626, −8.797275263769117, −8.139712771843271, −7.803720040113340, −7.232399414722928, −6.987825088465027, −6.147280573518408, −5.670181900071414, −5.272827550000158, −4.751132835733791, −3.874121667947406, −3.200980217451794, −2.724768734365289, −2.190533304301162, −1.659363963406450, −0.5537385711433303, 0,
0.5537385711433303, 1.659363963406450, 2.190533304301162, 2.724768734365289, 3.200980217451794, 3.874121667947406, 4.751132835733791, 5.272827550000158, 5.670181900071414, 6.147280573518408, 6.987825088465027, 7.232399414722928, 7.803720040113340, 8.139712771843271, 8.797275263769117, 9.529589595090626, 9.732162655585116, 10.12680594221381, 10.49221597204801, 11.01575041549171, 11.85223355255401, 12.10182805350026, 12.53781220120570, 13.09109149383375, 13.56770205675289
