| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 9-s + 4·12-s − 13-s − 4·16-s + 7·17-s + 2·18-s − 7·19-s − 5·25-s − 2·26-s − 4·27-s + 29-s − 6·31-s − 8·32-s + 14·34-s + 2·36-s − 7·37-s − 14·38-s − 2·39-s + 4·41-s − 7·43-s − 12·47-s − 8·48-s − 7·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 1/3·9-s + 1.15·12-s − 0.277·13-s − 16-s + 1.69·17-s + 0.471·18-s − 1.60·19-s − 25-s − 0.392·26-s − 0.769·27-s + 0.185·29-s − 1.07·31-s − 1.41·32-s + 2.40·34-s + 1/3·36-s − 1.15·37-s − 2.27·38-s − 0.320·39-s + 0.624·41-s − 1.06·43-s − 1.75·47-s − 1.15·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−16.08152669014102, −15.43948275261955, −14.77480032923587, −14.60215306045763, −14.24086208920861, −13.59133551065044, −12.93951572534557, −12.77657952347471, −11.95747333034535, −11.53168600611454, −10.76838318952597, −9.928327699226115, −9.543900468372470, −8.724380272159054, −8.195345580749124, −7.700866884057346, −6.794316684452415, −6.311216433166249, −5.364471917580524, −5.147616935356040, −3.995905282816201, −3.713968829217732, −3.094575838166054, −2.316558852264583, −1.715064242480141, 0,
1.715064242480141, 2.316558852264583, 3.094575838166054, 3.713968829217732, 3.995905282816201, 5.147616935356040, 5.364471917580524, 6.311216433166249, 6.794316684452415, 7.700866884057346, 8.195345580749124, 8.724380272159054, 9.543900468372470, 9.928327699226115, 10.76838318952597, 11.53168600611454, 11.95747333034535, 12.77657952347471, 12.93951572534557, 13.59133551065044, 14.24086208920861, 14.60215306045763, 14.77480032923587, 15.43948275261955, 16.08152669014102
