| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 12-s + 4·13-s − 2·14-s + 16-s + 17-s + 18-s − 2·19-s + 2·21-s − 2·23-s − 24-s + 4·26-s − 27-s − 2·28-s − 2·29-s − 4·31-s + 32-s + 34-s + 36-s − 6·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.417·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.986·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402101225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402101225\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.86837879835314, −12.18377571044926, −11.80936460687647, −11.42102856396028, −10.85371586893391, −10.35680228393163, −10.26641487657717, −9.446131070824093, −9.017823286922882, −8.553259455909675, −7.923233482820105, −7.406073228411602, −6.897844268596253, −6.459198184962204, −6.010479894671180, −5.687226448978951, −5.127247230742597, −4.564580321404101, −3.963710251582046, −3.570621590590256, −3.148440075594083, −2.408780856780819, −1.702411372568657, −1.257756761734172, −0.2841288951151797,
0.2841288951151797, 1.257756761734172, 1.702411372568657, 2.408780856780819, 3.148440075594083, 3.570621590590256, 3.963710251582046, 4.564580321404101, 5.127247230742597, 5.687226448978951, 6.010479894671180, 6.459198184962204, 6.897844268596253, 7.406073228411602, 7.923233482820105, 8.553259455909675, 9.017823286922882, 9.446131070824093, 10.26641487657717, 10.35680228393163, 10.85371586893391, 11.42102856396028, 11.80936460687647, 12.18377571044926, 12.86837879835314
