| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3·7-s − 8-s + 9-s − 3·11-s − 12-s + 13-s + 3·14-s + 16-s + 2·17-s − 18-s + 3·21-s + 3·22-s − 4·23-s + 24-s − 26-s − 27-s − 3·28-s + 10·29-s + 31-s − 32-s + 3·33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.654·21-s + 0.639·22-s − 0.834·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.566·28-s + 1.85·29-s + 0.179·31-s − 0.176·32-s + 0.522·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 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 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.972229178322202396417527504980, −7.24061871170606088819831761138, −6.46513946037455035213859743970, −5.98312328222860303879633802029, −5.17785411129813587456287280062, −4.15051595575684229069799281245, −3.15773327322568209826294856380, −2.41087241799822553469236564908, −1.05258675015093679696375135861, 0,
1.05258675015093679696375135861, 2.41087241799822553469236564908, 3.15773327322568209826294856380, 4.15051595575684229069799281245, 5.17785411129813587456287280062, 5.98312328222860303879633802029, 6.46513946037455035213859743970, 7.24061871170606088819831761138, 7.972229178322202396417527504980
