| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s + 15-s + 2·17-s + 4·19-s + 8·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s − 6·37-s − 2·41-s − 12·43-s − 45-s + 8·47-s − 7·49-s − 2·51-s − 2·53-s − 4·55-s − 4·57-s − 12·59-s − 2·61-s − 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.258·15-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.312·41-s − 1.82·43-s − 0.149·45-s + 1.16·47-s − 49-s − 0.280·51-s − 0.274·53-s − 0.539·55-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−16.02436239055061, −15.35312353556293, −14.83335484645282, −14.42127800418906, −13.66040027258962, −13.24281453805959, −12.38011843204605, −12.05039357124841, −11.64908031313788, −10.95515098721902, −10.54902238251947, −9.794044934120248, −9.160845516461425, −8.780492000893187, −7.963600438241483, −7.180301057404543, −6.949001266121128, −6.196318912659456, −5.494307397018733, −4.870318684199647, −4.319675692371293, −3.341283300656342, −3.107936683851166, −1.615969288561639, −1.160989273667117, 0,
1.160989273667117, 1.615969288561639, 3.107936683851166, 3.341283300656342, 4.319675692371293, 4.870318684199647, 5.494307397018733, 6.196318912659456, 6.949001266121128, 7.180301057404543, 7.963600438241483, 8.780492000893187, 9.160845516461425, 9.794044934120248, 10.54902238251947, 10.95515098721902, 11.64908031313788, 12.05039357124841, 12.38011843204605, 13.24281453805959, 13.66040027258962, 14.42127800418906, 14.83335484645282, 15.35312353556293, 16.02436239055061
