| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 11-s + 6·13-s − 15-s + 2·17-s + 4·19-s + 21-s + 8·23-s + 25-s − 27-s − 2·29-s + 8·31-s − 33-s − 35-s + 6·37-s − 6·39-s − 6·41-s − 4·43-s + 45-s + 8·47-s + 49-s − 2·51-s + 6·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.174·33-s − 0.169·35-s + 0.986·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.394953049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394953049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−7.58206047652752935863829788507, −6.88900511345019772205172709712, −6.29806278602476408627290687345, −5.73250782058675623593802806310, −5.11376358874663098579678366382, −4.23069026491416572693514249786, −3.42187944829232234157789623357, −2.75376870545636502526179038589, −1.39381530843462682949437086431, −0.889482462082146330610089155103,
0.889482462082146330610089155103, 1.39381530843462682949437086431, 2.75376870545636502526179038589, 3.42187944829232234157789623357, 4.23069026491416572693514249786, 5.11376358874663098579678366382, 5.73250782058675623593802806310, 6.29806278602476408627290687345, 6.88900511345019772205172709712, 7.58206047652752935863829788507
