| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·5-s + 2·6-s − 2·9-s − 4·10-s + 3·11-s − 2·12-s + 6·13-s − 2·15-s − 4·16-s − 2·17-s + 4·18-s − 6·19-s + 4·20-s − 6·22-s + 4·23-s − 25-s − 12·26-s + 5·27-s + 4·29-s + 4·30-s + 8·32-s − 3·33-s + 4·34-s − 4·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s + 0.816·6-s − 2/3·9-s − 1.26·10-s + 0.904·11-s − 0.577·12-s + 1.66·13-s − 0.516·15-s − 16-s − 0.485·17-s + 0.942·18-s − 1.37·19-s + 0.894·20-s − 1.27·22-s + 0.834·23-s − 1/5·25-s − 2.35·26-s + 0.962·27-s + 0.742·29-s + 0.730·30-s + 1.41·32-s − 0.522·33-s + 0.685·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.35117411663659, −13.94332288662429, −13.32953170211436, −13.11758329358275, −12.24122387916503, −11.65076092258699, −11.19564039353364, −10.80899456937381, −10.45947448522892, −9.764352692097021, −9.353778124986738, −8.729866245329580, −8.426122137425439, −8.187842744358089, −6.971117926173524, −6.645953086616255, −6.344761995048651, −5.689396545541571, −5.108573588038758, −4.283351010248584, −3.736157383240503, −2.774987275183602, −2.091693740443860, −1.431785718275064, −0.9055917252139673, 0,
0.9055917252139673, 1.431785718275064, 2.091693740443860, 2.774987275183602, 3.736157383240503, 4.283351010248584, 5.108573588038758, 5.689396545541571, 6.344761995048651, 6.645953086616255, 6.971117926173524, 8.187842744358089, 8.426122137425439, 8.729866245329580, 9.353778124986738, 9.764352692097021, 10.45947448522892, 10.80899456937381, 11.19564039353364, 11.65076092258699, 12.24122387916503, 13.11758329358275, 13.32953170211436, 13.94332288662429, 14.35117411663659
