| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 2·13-s + 15-s − 6·17-s + 4·19-s − 4·21-s + 25-s + 27-s − 6·29-s − 8·31-s − 4·35-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s + 45-s + 9·49-s − 6·51-s + 6·53-s + 4·57-s − 10·61-s − 4·63-s + 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s + 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 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 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.70437207815149, −13.28987815515262, −13.04988232108664, −12.60130573545532, −12.01589466025680, −11.33317805268743, −10.84320872021769, −10.42373606560807, −9.794152006041235, −9.312066369973248, −9.030665329626341, −8.747233949512238, −7.762048047779621, −7.379118467558885, −6.873266207529831, −6.329426810974199, −5.863140480262958, −5.389294800217086, −4.538965423897038, −3.962171622390928, −3.420512470683579, −3.005437167970143, −2.256500315137565, −1.796353526086149, −0.8231451536095404, 0,
0.8231451536095404, 1.796353526086149, 2.256500315137565, 3.005437167970143, 3.420512470683579, 3.962171622390928, 4.538965423897038, 5.389294800217086, 5.863140480262958, 6.329426810974199, 6.873266207529831, 7.379118467558885, 7.762048047779621, 8.747233949512238, 9.030665329626341, 9.312066369973248, 9.794152006041235, 10.42373606560807, 10.84320872021769, 11.33317805268743, 12.01589466025680, 12.60130573545532, 13.04988232108664, 13.28987815515262, 13.70437207815149
