| L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s − 2·12-s − 13-s + 4·16-s − 7·19-s − 21-s + 6·23-s + 27-s + 2·28-s + 6·29-s + 5·31-s − 2·36-s + 2·37-s − 39-s − 41-s − 7·43-s − 12·47-s + 4·48-s − 6·49-s + 2·52-s − 6·53-s − 7·57-s − 12·59-s − 13·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s + 16-s − 1.60·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 0.898·31-s − 1/3·36-s + 0.328·37-s − 0.160·39-s − 0.156·41-s − 1.06·43-s − 1.75·47-s + 0.577·48-s − 6/7·49-s + 0.277·52-s − 0.824·53-s − 0.927·57-s − 1.56·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−8.305945606391839205282292090836, −7.932329180005966885575360519907, −6.72329343913589780866860376462, −6.21989382792203032036374556840, −4.84399229198312468911478750066, −4.61230029625516083058344364246, −3.47311181004335955470223955657, −2.79504250888332980219546612739, −1.44680368185532689738807678075, 0,
1.44680368185532689738807678075, 2.79504250888332980219546612739, 3.47311181004335955470223955657, 4.61230029625516083058344364246, 4.84399229198312468911478750066, 6.21989382792203032036374556840, 6.72329343913589780866860376462, 7.932329180005966885575360519907, 8.305945606391839205282292090836
