| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 4·11-s − 15-s + 2·17-s + 4·19-s + 4·21-s + 25-s + 27-s − 10·29-s + 4·31-s + 4·33-s − 4·35-s − 10·37-s − 2·41-s + 4·43-s − 45-s − 8·47-s + 9·49-s + 2·51-s − 2·53-s − 4·55-s + 4·57-s + 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.696·33-s − 0.676·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.927898806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.927898806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.23203388087442, −12.90389485259096, −12.10188745889976, −11.80544549595621, −11.44077417877020, −11.08273625764425, −10.37963414087780, −9.868177039683620, −9.381704693246664, −8.828262307135347, −8.434553003571329, −8.015725742178591, −7.449861004297624, −7.132290672667551, −6.566364577648306, −5.756179029960649, −5.245091882484470, −4.824508051063875, −4.169580825669497, −3.612449008757296, −3.375307319771311, −2.366879717302456, −1.794067898259685, −1.335550426996025, −0.6515013460831608,
0.6515013460831608, 1.335550426996025, 1.794067898259685, 2.366879717302456, 3.375307319771311, 3.612449008757296, 4.169580825669497, 4.824508051063875, 5.245091882484470, 5.756179029960649, 6.566364577648306, 7.132290672667551, 7.449861004297624, 8.015725742178591, 8.434553003571329, 8.828262307135347, 9.381704693246664, 9.868177039683620, 10.37963414087780, 11.08273625764425, 11.44077417877020, 11.80544549595621, 12.10188745889976, 12.90389485259096, 13.23203388087442
