| L(s) = 1 | + 2-s + 4-s + 3·5-s − 2·7-s + 8-s + 3·10-s + 5·11-s − 2·13-s − 2·14-s + 16-s + 4·17-s + 3·20-s + 5·22-s − 23-s + 4·25-s − 2·26-s − 2·28-s − 3·29-s − 8·31-s + 32-s + 4·34-s − 6·35-s + 4·37-s + 3·40-s + 2·41-s + 5·43-s + 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s + 0.353·8-s + 0.948·10-s + 1.50·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.670·20-s + 1.06·22-s − 0.208·23-s + 4/5·25-s − 0.392·26-s − 0.377·28-s − 0.557·29-s − 1.43·31-s + 0.176·32-s + 0.685·34-s − 1.01·35-s + 0.657·37-s + 0.474·40-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
| \(L(1)\) |
\(\approx\) |
\(6.823376980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.823376980\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−13.49195240655743, −12.86000351642381, −12.39253623002826, −12.17927369621524, −11.54907519578007, −10.94967249719956, −10.46670289223390, −9.932328493714863, −9.466655492060887, −9.221020841122666, −8.751141858742066, −7.741966380308440, −7.363931557437999, −6.853292904844087, −6.131984997022388, −6.065431134566557, −5.499044952066133, −4.962535512725476, −4.209708861025025, −3.664959306045235, −3.306721250779607, −2.361111485465671, −2.130308640533131, −1.322406266001268, −0.6935385588591620,
0.6935385588591620, 1.322406266001268, 2.130308640533131, 2.361111485465671, 3.306721250779607, 3.664959306045235, 4.209708861025025, 4.962535512725476, 5.499044952066133, 6.065431134566557, 6.131984997022388, 6.853292904844087, 7.363931557437999, 7.741966380308440, 8.751141858742066, 9.221020841122666, 9.466655492060887, 9.932328493714863, 10.46670289223390, 10.94967249719956, 11.54907519578007, 12.17927369621524, 12.39253623002826, 12.86000351642381, 13.49195240655743
