| L(s) = 1 | + 2·7-s − 11-s + 2·13-s − 6·17-s + 4·19-s − 2·29-s − 8·31-s + 2·41-s + 2·43-s − 8·47-s − 3·49-s + 8·53-s + 8·59-s + 10·61-s + 8·67-s + 12·71-s + 14·73-s − 2·77-s + 8·79-s + 6·83-s + 14·89-s + 4·91-s + 12·97-s − 14·101-s − 4·103-s − 10·107-s − 10·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.371·29-s − 1.43·31-s + 0.312·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 1.09·53-s + 1.04·59-s + 1.28·61-s + 0.977·67-s + 1.42·71-s + 1.63·73-s − 0.227·77-s + 0.900·79-s + 0.658·83-s + 1.48·89-s + 0.419·91-s + 1.21·97-s − 1.39·101-s − 0.394·103-s − 0.966·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.108235086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108235086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.81638744935393894343716970540, −6.89977129614167402971097381912, −6.46306017445929781479128063412, −5.26741619056406933449330257279, −5.22255274044825131097680886671, −4.08583574209629152834905756204, −3.57524335569119375218287770700, −2.43083152265515410842422616071, −1.81029113901524102004924270340, −0.68889155267163505518912894043,
0.68889155267163505518912894043, 1.81029113901524102004924270340, 2.43083152265515410842422616071, 3.57524335569119375218287770700, 4.08583574209629152834905756204, 5.22255274044825131097680886671, 5.26741619056406933449330257279, 6.46306017445929781479128063412, 6.89977129614167402971097381912, 7.81638744935393894343716970540
