| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 7·13-s − 15-s + 6·17-s − 3·19-s + 2·23-s + 25-s + 27-s + 2·29-s + 7·31-s − 4·33-s + 7·37-s + 7·39-s + 8·41-s + 5·43-s − 45-s + 10·47-s + 6·51-s + 8·53-s + 4·55-s − 3·57-s − 10·59-s − 6·61-s − 7·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.94·13-s − 0.258·15-s + 1.45·17-s − 0.688·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.25·31-s − 0.696·33-s + 1.15·37-s + 1.12·39-s + 1.24·41-s + 0.762·43-s − 0.149·45-s + 1.45·47-s + 0.840·51-s + 1.09·53-s + 0.539·55-s − 0.397·57-s − 1.30·59-s − 0.768·61-s − 0.868·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.594910346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.594910346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.49749038518138, −14.10525760843602, −13.55715579290664, −13.05800054349714, −12.69429659726737, −12.07547972827018, −11.45254954464260, −10.87474461444349, −10.43132370002505, −10.07459543061583, −9.145774272969032, −8.811858149415047, −8.178733563300799, −7.799682418349827, −7.397929561028747, −6.530473733117940, −5.852371359215517, −5.618737918000685, −4.448392656506360, −4.283892120359106, −3.363082356488797, −2.958618386364970, −2.307320427535074, −1.215634402896517, −0.7487034569804905,
0.7487034569804905, 1.215634402896517, 2.307320427535074, 2.958618386364970, 3.363082356488797, 4.283892120359106, 4.448392656506360, 5.618737918000685, 5.852371359215517, 6.530473733117940, 7.397929561028747, 7.799682418349827, 8.178733563300799, 8.811858149415047, 9.145774272969032, 10.07459543061583, 10.43132370002505, 10.87474461444349, 11.45254954464260, 12.07547972827018, 12.69429659726737, 13.05800054349714, 13.55715579290664, 14.10525760843602, 14.49749038518138
