| L(s) = 1 | + 4·5-s − 3·9-s + 4·11-s + 4·13-s + 6·19-s + 4·23-s + 11·25-s + 6·29-s + 2·31-s + 37-s + 6·41-s − 4·43-s − 12·45-s − 12·47-s − 10·53-s + 16·55-s + 10·59-s − 8·61-s + 16·65-s − 4·67-s − 2·73-s − 4·79-s + 9·81-s − 16·89-s + 24·95-s − 4·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 9-s + 1.20·11-s + 1.10·13-s + 1.37·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s + 0.359·31-s + 0.164·37-s + 0.937·41-s − 0.609·43-s − 1.78·45-s − 1.75·47-s − 1.37·53-s + 2.15·55-s + 1.30·59-s − 1.02·61-s + 1.98·65-s − 0.488·67-s − 0.234·73-s − 0.450·79-s + 81-s − 1.69·89-s + 2.46·95-s − 0.406·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.499328496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.499328496\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.63416165291692, −13.33616442300684, −12.70498202039085, −12.19136598490490, −11.41473253907171, −11.33515670349899, −10.72170863413852, −9.948959695528666, −9.758528007880448, −9.211059526403853, −8.711259175119670, −8.472774762706967, −7.647954552932088, −6.864897151710359, −6.468578760105902, −6.027548384632341, −5.705687201182117, −5.003623315732421, −4.614590433131640, −3.613453429514603, −3.067979557018565, −2.719955043623721, −1.757678712073319, −1.331313931850530, −0.7721308456465888,
0.7721308456465888, 1.331313931850530, 1.757678712073319, 2.719955043623721, 3.067979557018565, 3.613453429514603, 4.614590433131640, 5.003623315732421, 5.705687201182117, 6.027548384632341, 6.468578760105902, 6.864897151710359, 7.647954552932088, 8.472774762706967, 8.711259175119670, 9.211059526403853, 9.758528007880448, 9.948959695528666, 10.72170863413852, 11.33515670349899, 11.41473253907171, 12.19136598490490, 12.70498202039085, 13.33616442300684, 13.63416165291692
