| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s − 4·13-s + 16-s − 18-s − 8·19-s + 22-s − 6·23-s − 24-s + 4·26-s + 27-s + 6·29-s + 4·31-s − 32-s − 33-s + 36-s − 8·37-s + 8·38-s − 4·39-s − 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.174·33-s + 1/6·36-s − 1.31·37-s + 1.29·38-s − 0.640·39-s − 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.31002071188777, −13.80383116068818, −13.29964279756915, −12.52316992121720, −12.28338920447862, −11.90437552230824, −11.03821910730956, −10.57865362747999, −10.16458695884558, −9.768023629300971, −9.207647280412237, −8.471361748406100, −8.311882259591677, −7.832197511193266, −7.065825574845927, −6.704414651340833, −6.209376486019220, −5.345005262110409, −4.878572730481551, −4.052362921276137, −3.714542476172970, −2.605614613144412, −2.402883737525071, −1.825748074976116, −0.7905362873623260, 0,
0.7905362873623260, 1.825748074976116, 2.402883737525071, 2.605614613144412, 3.714542476172970, 4.052362921276137, 4.878572730481551, 5.345005262110409, 6.209376486019220, 6.704414651340833, 7.065825574845927, 7.832197511193266, 8.311882259591677, 8.471361748406100, 9.207647280412237, 9.768023629300971, 10.16458695884558, 10.57865362747999, 11.03821910730956, 11.90437552230824, 12.28338920447862, 12.52316992121720, 13.29964279756915, 13.80383116068818, 14.31002071188777
