| L(s) = 1 | + 3-s − 2·5-s + 9-s + 13-s − 2·15-s + 2·17-s − 6·19-s − 8·23-s − 25-s + 27-s + 4·29-s − 4·31-s + 6·37-s + 39-s + 8·43-s − 2·45-s + 2·51-s − 6·57-s − 8·59-s + 10·61-s − 2·65-s + 14·67-s − 8·69-s − 8·71-s + 2·73-s − 75-s + 14·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 0.485·17-s − 1.37·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.718·31-s + 0.986·37-s + 0.160·39-s + 1.21·43-s − 0.298·45-s + 0.280·51-s − 0.794·57-s − 1.04·59-s + 1.28·61-s − 0.248·65-s + 1.71·67-s − 0.963·69-s − 0.949·71-s + 0.234·73-s − 0.115·75-s + 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.09609441265682, −12.53270454726573, −12.23829366810116, −11.85622440558323, −11.09095582237624, −10.94672134715792, −10.31700722922709, −9.760693771747415, −9.468305585062240, −8.721564431674801, −8.373915177817704, −7.930780756909784, −7.680887754760324, −7.017488727937020, −6.482150335370605, −5.988478000171709, −5.507476003998260, −4.680005763951008, −4.225162327315892, −3.846345022998999, −3.464154286543794, −2.593041406250694, −2.246799486739953, −1.534845218062358, −0.7249467857494007, 0,
0.7249467857494007, 1.534845218062358, 2.246799486739953, 2.593041406250694, 3.464154286543794, 3.846345022998999, 4.225162327315892, 4.680005763951008, 5.507476003998260, 5.988478000171709, 6.482150335370605, 7.017488727937020, 7.680887754760324, 7.930780756909784, 8.373915177817704, 8.721564431674801, 9.468305585062240, 9.760693771747415, 10.31700722922709, 10.94672134715792, 11.09095582237624, 11.85622440558323, 12.23829366810116, 12.53270454726573, 13.09609441265682
