| L(s) = 1 | + 3-s − 2·5-s + 4·7-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 6·17-s − 4·19-s + 4·21-s − 25-s + 27-s − 2·29-s − 4·31-s + 4·33-s − 8·35-s + 2·37-s + 2·39-s + 2·41-s + 4·43-s − 2·45-s − 8·47-s + 9·49-s − 6·51-s − 10·53-s − 8·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s − 1.35·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s − 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.415737208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415737208\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 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.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.46429529504907365579234739579, −11.31959077603206060511694459208, −11.01151255843213055292894337270, −9.240123327709471504547415939693, −8.460175078145674768211752918036, −7.67780945046588747720337476950, −6.44896944953500655110243120870, −4.62570703678849832502580328887, −3.85566973953101374011358603281, −1.83689405023894510739841616557,
1.83689405023894510739841616557, 3.85566973953101374011358603281, 4.62570703678849832502580328887, 6.44896944953500655110243120870, 7.67780945046588747720337476950, 8.460175078145674768211752918036, 9.240123327709471504547415939693, 11.01151255843213055292894337270, 11.31959077603206060511694459208, 12.46429529504907365579234739579
