| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s + 5·11-s − 4·16-s + 2·17-s + 6·19-s + 2·20-s + 10·22-s + 6·23-s + 25-s − 2·29-s + 8·31-s − 8·32-s + 4·34-s − 10·37-s + 12·38-s + 10·43-s + 10·44-s + 12·46-s + 8·47-s + 2·50-s − 2·53-s + 5·55-s − 4·58-s + 6·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 1.50·11-s − 16-s + 0.485·17-s + 1.37·19-s + 0.447·20-s + 2.13·22-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 1.41·32-s + 0.685·34-s − 1.64·37-s + 1.94·38-s + 1.52·43-s + 1.50·44-s + 1.76·46-s + 1.16·47-s + 0.282·50-s − 0.274·53-s + 0.674·55-s − 0.525·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.53517692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.53517692\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−12.56520121241487, −12.06544049913393, −11.79606542329752, −11.37108406792245, −10.88181039703593, −10.28887574533784, −9.759503243146492, −9.291909218361665, −8.929830788515081, −8.572490736620181, −7.660294379216448, −7.259038849433477, −6.787090824225944, −6.370928056470167, −5.896372531343462, −5.368427236225814, −5.089041190214008, −4.470473046065535, −3.970975180703651, −3.518666205131558, −3.039150839258093, −2.573744292783081, −1.833523628323901, −1.161542438410844, −0.7099782177325375,
0.7099782177325375, 1.161542438410844, 1.833523628323901, 2.573744292783081, 3.039150839258093, 3.518666205131558, 3.970975180703651, 4.470473046065535, 5.089041190214008, 5.368427236225814, 5.896372531343462, 6.370928056470167, 6.787090824225944, 7.259038849433477, 7.660294379216448, 8.572490736620181, 8.929830788515081, 9.291909218361665, 9.759503243146492, 10.28887574533784, 10.88181039703593, 11.37108406792245, 11.79606542329752, 12.06544049913393, 12.56520121241487
