| L(s) = 1 | − 5-s + 7-s − 3·9-s + 6·11-s + 2·13-s − 3·17-s + 6·19-s + 23-s + 25-s − 3·29-s − 3·31-s − 35-s − 37-s + 9·41-s + 8·43-s + 3·45-s + 4·47-s − 6·49-s − 53-s − 6·55-s − 59-s − 8·61-s − 3·63-s − 2·65-s + 7·67-s − 5·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s + 1.80·11-s + 0.554·13-s − 0.727·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s − 0.538·31-s − 0.169·35-s − 0.164·37-s + 1.40·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s − 6/7·49-s − 0.137·53-s − 0.809·55-s − 0.130·59-s − 1.02·61-s − 0.377·63-s − 0.248·65-s + 0.855·67-s − 0.593·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.079027754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079027754\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.82475827235358411092181433569, −7.30985720819668641348214819360, −6.42902291644534422762585258760, −5.91348077736240374383411736283, −5.09357703180522838251878256848, −4.18978636845118664263387616696, −3.64142320448801410323297387172, −2.81760492012154987874746453886, −1.67265854460187075003413542223, −0.76222428069584049137434436993,
0.76222428069584049137434436993, 1.67265854460187075003413542223, 2.81760492012154987874746453886, 3.64142320448801410323297387172, 4.18978636845118664263387616696, 5.09357703180522838251878256848, 5.91348077736240374383411736283, 6.42902291644534422762585258760, 7.30985720819668641348214819360, 7.82475827235358411092181433569
