| L(s) = 1 | − 2·5-s + 7-s + 4·13-s + 19-s + 6·23-s − 25-s − 3·29-s − 9·31-s − 2·35-s + 6·37-s + 6·41-s + 6·43-s + 47-s + 49-s − 13·53-s − 7·59-s − 10·61-s − 8·65-s + 6·67-s − 6·71-s + 2·73-s + 16·79-s − 15·83-s + 14·89-s + 4·91-s − 2·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.10·13-s + 0.229·19-s + 1.25·23-s − 1/5·25-s − 0.557·29-s − 1.61·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.914·43-s + 0.145·47-s + 1/7·49-s − 1.78·53-s − 0.911·59-s − 1.28·61-s − 0.992·65-s + 0.733·67-s − 0.712·71-s + 0.234·73-s + 1.80·79-s − 1.64·83-s + 1.48·89-s + 0.419·91-s − 0.205·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−14.38421856845493, −13.91586028657373, −13.24130678201450, −12.84688173165602, −12.41788925168622, −11.72887934474984, −11.26234617971697, −10.88930717825139, −10.68001406404712, −9.639065784818877, −9.129044368678940, −8.922496184090742, −7.998244103095949, −7.767731995001447, −7.327610159663109, −6.593612057905397, −5.989064213083539, −5.513840710681007, −4.751306527739299, −4.288811008694345, −3.614424503838009, −3.254735465923983, −2.401296577308863, −1.555808614686972, −0.9375658986948882, 0,
0.9375658986948882, 1.555808614686972, 2.401296577308863, 3.254735465923983, 3.614424503838009, 4.288811008694345, 4.751306527739299, 5.513840710681007, 5.989064213083539, 6.593612057905397, 7.327610159663109, 7.767731995001447, 7.998244103095949, 8.922496184090742, 9.129044368678940, 9.639065784818877, 10.68001406404712, 10.88930717825139, 11.26234617971697, 11.72887934474984, 12.41788925168622, 12.84688173165602, 13.24130678201450, 13.91586028657373, 14.38421856845493
