| L(s) = 1 | + 3-s + 3·5-s − 5·7-s − 2·9-s + 4·13-s + 3·15-s − 6·17-s + 19-s − 5·21-s + 2·23-s + 4·25-s − 5·27-s + 5·29-s − 10·31-s − 15·35-s + 2·37-s + 4·39-s − 9·41-s − 6·43-s − 6·45-s + 4·47-s + 18·49-s − 6·51-s + 3·53-s + 57-s + 59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 1.88·7-s − 2/3·9-s + 1.10·13-s + 0.774·15-s − 1.45·17-s + 0.229·19-s − 1.09·21-s + 0.417·23-s + 4/5·25-s − 0.962·27-s + 0.928·29-s − 1.79·31-s − 2.53·35-s + 0.328·37-s + 0.640·39-s − 1.40·41-s − 0.914·43-s − 0.894·45-s + 0.583·47-s + 18/7·49-s − 0.840·51-s + 0.412·53-s + 0.132·57-s + 0.130·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 59 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.552762129303510412992353611593, −7.16011760331644340864374355931, −6.55743353998257394228295643521, −5.99325606085737183763065707255, −5.42457561474262890298477846896, −4.07274885625562226430602271382, −3.19435721478314527164485870603, −2.66641344801336135028410261450, −1.65473366283942033394093775383, 0,
1.65473366283942033394093775383, 2.66641344801336135028410261450, 3.19435721478314527164485870603, 4.07274885625562226430602271382, 5.42457561474262890298477846896, 5.99325606085737183763065707255, 6.55743353998257394228295643521, 7.16011760331644340864374355931, 8.552762129303510412992353611593
