| L(s) = 1 | + 2·3-s − 5-s + 4·7-s + 9-s − 4·11-s + 4·13-s − 2·15-s − 2·17-s + 19-s + 8·21-s + 25-s − 4·27-s − 2·29-s + 8·31-s − 8·33-s − 4·35-s − 4·37-s + 8·39-s + 6·41-s − 45-s + 12·47-s + 9·49-s − 4·51-s + 8·53-s + 4·55-s + 2·57-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.516·15-s − 0.485·17-s + 0.229·19-s + 1.74·21-s + 1/5·25-s − 0.769·27-s − 0.371·29-s + 1.43·31-s − 1.39·33-s − 0.676·35-s − 0.657·37-s + 1.28·39-s + 0.937·41-s − 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.560·51-s + 1.09·53-s + 0.539·55-s + 0.264·57-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.321842416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.321842416\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.957897188736796047407716030057, −7.83204414349715077231995955353, −6.90838586097089125721138961188, −5.79088931298754069749598117310, −5.12692209148964588109531291237, −4.29593188524500848954459800401, −3.64891704934951363544948150911, −2.65543894686412903871416838458, −2.08558235281628555403017728095, −0.926641843009645575190439776976,
0.926641843009645575190439776976, 2.08558235281628555403017728095, 2.65543894686412903871416838458, 3.64891704934951363544948150911, 4.29593188524500848954459800401, 5.12692209148964588109531291237, 5.79088931298754069749598117310, 6.90838586097089125721138961188, 7.83204414349715077231995955353, 7.957897188736796047407716030057
