| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s + 2·11-s − 2·12-s + 6·13-s + 16-s − 3·17-s + 18-s + 2·19-s + 2·22-s + 6·23-s − 2·24-s + 6·26-s + 4·27-s − 9·29-s + 10·31-s + 32-s − 4·33-s − 3·34-s + 36-s + 2·38-s − 12·39-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1.66·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s + 0.426·22-s + 1.25·23-s − 0.408·24-s + 1.17·26-s + 0.769·27-s − 1.67·29-s + 1.79·31-s + 0.176·32-s − 0.696·33-s − 0.514·34-s + 1/6·36-s + 0.324·38-s − 1.92·39-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.419958820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.419958820\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−14.05615722039347, −13.60056502944597, −13.08976307017294, −12.75663064956909, −12.05508882399126, −11.59893581950728, −11.14589118806428, −10.99829982593482, −10.42230448454168, −9.603002685624831, −9.126644707808147, −8.495075914197412, −7.992720740003034, −7.062432068831241, −6.763652229939938, −6.223648730252060, −5.723305295239297, −5.381887943098875, −4.557650676853449, −4.210802574596755, −3.468810596297231, −2.925679680937153, −2.008238031353943, −1.159313267144358, −0.6826939437498880,
0.6826939437498880, 1.159313267144358, 2.008238031353943, 2.925679680937153, 3.468810596297231, 4.210802574596755, 4.557650676853449, 5.381887943098875, 5.723305295239297, 6.223648730252060, 6.763652229939938, 7.062432068831241, 7.992720740003034, 8.495075914197412, 9.126644707808147, 9.603002685624831, 10.42230448454168, 10.99829982593482, 11.14589118806428, 11.59893581950728, 12.05508882399126, 12.75663064956909, 13.08976307017294, 13.60056502944597, 14.05615722039347
