| L(s) = 1 | − 3-s + 3·5-s + 9-s − 3·15-s − 17-s − 4·23-s + 4·25-s − 27-s + 3·29-s − 8·31-s − 5·37-s + 3·41-s − 4·43-s + 3·45-s + 8·47-s − 7·49-s + 51-s − 13·53-s − 12·59-s + 15·61-s − 12·67-s + 4·69-s − 8·71-s + 3·73-s − 4·75-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.774·15-s − 0.242·17-s − 0.834·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s − 1.43·31-s − 0.821·37-s + 0.468·41-s − 0.609·43-s + 0.447·45-s + 1.16·47-s − 49-s + 0.140·51-s − 1.78·53-s − 1.56·59-s + 1.92·61-s − 1.46·67-s + 0.481·69-s − 0.949·71-s + 0.351·73-s − 0.461·75-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 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
−7.35806729137958711531664902334, −6.58501106616391728562045756115, −6.07589780882340551764586811195, −5.48921778277559217286886482591, −4.85935508655331033288333768424, −3.99558745409722001592661650199, −2.99830302876158519076817978513, −2.02425413745695052632367493481, −1.42900936833239416444795719221, 0,
1.42900936833239416444795719221, 2.02425413745695052632367493481, 2.99830302876158519076817978513, 3.99558745409722001592661650199, 4.85935508655331033288333768424, 5.48921778277559217286886482591, 6.07589780882340551764586811195, 6.58501106616391728562045756115, 7.35806729137958711531664902334
