| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 2·13-s − 14-s + 16-s + 2·17-s + 20-s − 4·23-s − 4·25-s + 2·26-s + 28-s + 4·29-s − 2·31-s − 32-s − 2·34-s + 35-s + 7·37-s − 40-s − 5·41-s − 4·43-s + 4·46-s − 11·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.223·20-s − 0.834·23-s − 4/5·25-s + 0.392·26-s + 0.188·28-s + 0.742·29-s − 0.359·31-s − 0.176·32-s − 0.342·34-s + 0.169·35-s + 1.15·37-s − 0.158·40-s − 0.780·41-s − 0.609·43-s + 0.589·46-s − 1.60·47-s + 1/7·49-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 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.89907150976385, −14.47092384490036, −13.85836617841041, −13.43142169984718, −12.72322712196469, −12.22887094305103, −11.66115643690950, −11.31162319574671, −10.59558219708792, −10.04460968567591, −9.697930098871461, −9.294498883822590, −8.392305494351047, −8.087427769742766, −7.647262478357032, −6.826427642066116, −6.444993384709929, −5.736282609566409, −5.200364614573230, −4.564735080911106, −3.754147961433622, −3.085103709364627, −2.243840956610988, −1.817350757535466, −0.9616803894799181, 0,
0.9616803894799181, 1.817350757535466, 2.243840956610988, 3.085103709364627, 3.754147961433622, 4.564735080911106, 5.200364614573230, 5.736282609566409, 6.444993384709929, 6.826427642066116, 7.647262478357032, 8.087427769742766, 8.392305494351047, 9.294498883822590, 9.697930098871461, 10.04460968567591, 10.59558219708792, 11.31162319574671, 11.66115643690950, 12.22887094305103, 12.72322712196469, 13.43142169984718, 13.85836617841041, 14.47092384490036, 14.89907150976385
