| L(s) = 1 | − 7-s − 5·13-s + 7·19-s − 5·25-s − 4·31-s − 11·37-s − 8·43-s − 6·49-s + 61-s − 5·67-s − 7·73-s + 17·79-s + 5·91-s − 19·97-s − 13·103-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.38·13-s + 1.60·19-s − 25-s − 0.718·31-s − 1.80·37-s − 1.21·43-s − 6/7·49-s + 0.128·61-s − 0.610·67-s − 0.819·73-s + 1.91·79-s + 0.524·91-s − 1.92·97-s − 1.28·103-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−9.093756787868480250299391378087, −8.013308166903493545899536406476, −7.33966183117961392379729992729, −6.66054107823412239247003697818, −5.50710978221504476175964668301, −4.97480875199317599127464128318, −3.74735768991078116659064649628, −2.90681031253330114231046316229, −1.71126395023482908495018202224, 0,
1.71126395023482908495018202224, 2.90681031253330114231046316229, 3.74735768991078116659064649628, 4.97480875199317599127464128318, 5.50710978221504476175964668301, 6.66054107823412239247003697818, 7.33966183117961392379729992729, 8.013308166903493545899536406476, 9.093756787868480250299391378087
