| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 3·11-s − 6·13-s − 15-s − 4·17-s − 19-s + 21-s + 3·23-s + 25-s + 27-s + 8·29-s + 31-s + 3·33-s − 35-s − 12·37-s − 6·39-s − 9·43-s − 45-s + 2·47-s − 6·49-s − 4·51-s − 53-s − 3·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.66·13-s − 0.258·15-s − 0.970·17-s − 0.229·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s + 0.179·31-s + 0.522·33-s − 0.169·35-s − 1.97·37-s − 0.960·39-s − 1.37·43-s − 0.149·45-s + 0.291·47-s − 6/7·49-s − 0.560·51-s − 0.137·53-s − 0.404·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.51307330127334681623695326989, −6.88737159726099618840894684259, −6.48935354584100209972404930340, −5.10183612013785584425561585854, −4.74401750017326714183781191498, −3.96014365990342157329614922367, −3.08871100539544499737162747792, −2.31533490436427976694050305447, −1.40461946060936575756399815851, 0,
1.40461946060936575756399815851, 2.31533490436427976694050305447, 3.08871100539544499737162747792, 3.96014365990342157329614922367, 4.74401750017326714183781191498, 5.10183612013785584425561585854, 6.48935354584100209972404930340, 6.88737159726099618840894684259, 7.51307330127334681623695326989
