| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 7-s + 4·10-s + 11-s − 5·13-s − 2·14-s − 4·16-s − 2·17-s + 3·19-s − 4·20-s − 2·22-s − 4·23-s − 25-s + 10·26-s + 2·28-s − 6·29-s − 8·31-s + 8·32-s + 4·34-s − 2·35-s − 9·37-s − 6·38-s + 4·41-s + 2·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 0.377·7-s + 1.26·10-s + 0.301·11-s − 1.38·13-s − 0.534·14-s − 16-s − 0.485·17-s + 0.688·19-s − 0.894·20-s − 0.426·22-s − 0.834·23-s − 1/5·25-s + 1.96·26-s + 0.377·28-s − 1.11·29-s − 1.43·31-s + 1.41·32-s + 0.685·34-s − 0.338·35-s − 1.47·37-s − 0.973·38-s + 0.624·41-s + 0.301·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−11.26965922032386947247401320935, −10.12591225010903813770667990591, −9.396520277869346801680336984619, −8.416888851848471678140027674283, −7.59139552718632210907564226259, −6.99610532434032137756158317446, −5.19396563839291084458776836793, −3.88790768469975700348723044941, −1.98678025944303973088896542660, 0,
1.98678025944303973088896542660, 3.88790768469975700348723044941, 5.19396563839291084458776836793, 6.99610532434032137756158317446, 7.59139552718632210907564226259, 8.416888851848471678140027674283, 9.396520277869346801680336984619, 10.12591225010903813770667990591, 11.26965922032386947247401320935
