| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 6·11-s − 4·13-s − 4·15-s + 2·17-s + 4·19-s + 23-s − 25-s − 4·27-s + 10·29-s + 8·31-s − 12·33-s + 8·37-s − 8·39-s + 2·41-s − 6·43-s − 2·45-s − 12·47-s + 4·51-s − 12·53-s + 12·55-s + 8·57-s − 6·59-s − 6·61-s + 8·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.80·11-s − 1.10·13-s − 1.03·15-s + 0.485·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.769·27-s + 1.85·29-s + 1.43·31-s − 2.08·33-s + 1.31·37-s − 1.28·39-s + 0.312·41-s − 0.914·43-s − 0.298·45-s − 1.75·47-s + 0.560·51-s − 1.64·53-s + 1.61·55-s + 1.05·57-s − 0.781·59-s − 0.768·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.40254405233161, −13.79858110319216, −13.54324973467546, −12.87978679932791, −12.39931303461083, −11.92987557882958, −11.40180059497039, −10.85673360682524, −10.06004897451817, −9.863580154030336, −9.380187514529012, −8.508834974181571, −8.043052849693840, −7.836106885660436, −7.574264137122896, −6.705920676783484, −6.113664567356015, −5.161406762299691, −4.879411814766005, −4.336302706015150, −3.369509304123335, −2.962741973797259, −2.703728964634201, −1.907475300426016, −0.8242761457370348, 0,
0.8242761457370348, 1.907475300426016, 2.703728964634201, 2.962741973797259, 3.369509304123335, 4.336302706015150, 4.879411814766005, 5.161406762299691, 6.113664567356015, 6.705920676783484, 7.574264137122896, 7.836106885660436, 8.043052849693840, 8.508834974181571, 9.380187514529012, 9.863580154030336, 10.06004897451817, 10.85673360682524, 11.40180059497039, 11.92987557882958, 12.39931303461083, 12.87978679932791, 13.54324973467546, 13.79858110319216, 14.40254405233161
