| L(s) = 1 | + 3-s − 2·5-s + 2·7-s + 9-s + 6·11-s − 2·13-s − 2·15-s + 2·21-s + 23-s − 25-s + 27-s + 6·29-s − 8·31-s + 6·33-s − 4·35-s − 2·39-s + 10·41-s + 12·43-s − 2·45-s + 8·47-s − 3·49-s + 2·53-s − 12·55-s + 12·59-s + 4·61-s + 2·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.516·15-s + 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.676·35-s − 0.320·39-s + 1.56·41-s + 1.82·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 1.61·55-s + 1.56·59-s + 0.512·61-s + 0.251·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014995243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014995243\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.595984735045838060267723644418, −9.006150174861512066746877136440, −8.206972512375746589573867040310, −7.43829407920294250781299014536, −6.76482292412091819449305273660, −5.51909301075206075300511483684, −4.21717016341822456956547144558, −3.92491062937506799522122153513, −2.49907678412809416629747058134, −1.15578668685740236008003655593,
1.15578668685740236008003655593, 2.49907678412809416629747058134, 3.92491062937506799522122153513, 4.21717016341822456956547144558, 5.51909301075206075300511483684, 6.76482292412091819449305273660, 7.43829407920294250781299014536, 8.206972512375746589573867040310, 9.006150174861512066746877136440, 9.595984735045838060267723644418
