| L(s) = 1 | + 3-s − 4·5-s + 4·7-s + 9-s − 13-s − 4·15-s + 4·17-s + 4·21-s − 8·23-s + 11·25-s + 27-s + 8·29-s + 2·31-s − 16·35-s + 2·37-s − 39-s + 10·41-s + 6·43-s − 4·45-s + 9·49-s + 4·51-s − 6·53-s + 14·61-s + 4·63-s + 4·65-s − 6·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 1.03·15-s + 0.970·17-s + 0.872·21-s − 1.66·23-s + 11/5·25-s + 0.192·27-s + 1.48·29-s + 0.359·31-s − 2.70·35-s + 0.328·37-s − 0.160·39-s + 1.56·41-s + 0.914·43-s − 0.596·45-s + 9/7·49-s + 0.560·51-s − 0.824·53-s + 1.79·61-s + 0.503·63-s + 0.496·65-s − 0.733·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.79827523988668, −12.98335268829243, −12.40640684857077, −12.11267959461423, −11.73543475666558, −11.25738971122625, −10.86976489723125, −10.19447213834087, −9.863957117995558, −9.034200197009619, −8.447156322380037, −8.222759471415717, −7.821890096643498, −7.446169957264655, −7.055146783992792, −6.146850412924758, −5.623222917272189, −4.801739321401551, −4.463792703523461, −4.095455892523674, −3.554317653831100, −2.788817797995863, −2.380901493248486, −1.376859869976055, −0.9631516755095550, 0,
0.9631516755095550, 1.376859869976055, 2.380901493248486, 2.788817797995863, 3.554317653831100, 4.095455892523674, 4.463792703523461, 4.801739321401551, 5.623222917272189, 6.146850412924758, 7.055146783992792, 7.446169957264655, 7.821890096643498, 8.222759471415717, 8.447156322380037, 9.034200197009619, 9.863957117995558, 10.19447213834087, 10.86976489723125, 11.25738971122625, 11.73543475666558, 12.11267959461423, 12.40640684857077, 12.98335268829243, 13.79827523988668
