| L(s) = 1 | + 5·7-s + 8·19-s − 5·25-s + 11·31-s + 10·37-s − 5·43-s + 18·49-s − 61-s + 5·67-s + 7·73-s − 17·79-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 1.83·19-s − 25-s + 1.97·31-s + 1.64·37-s − 0.762·43-s + 18/7·49-s − 0.128·61-s + 0.610·67-s + 0.819·73-s − 1.91·79-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.637940330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.637940330\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−15.40398044100071, −14.82275895666786, −14.29789066608751, −13.82367783706395, −13.51047356691001, −12.67268660871602, −11.85450981447881, −11.56846370333840, −11.36524100348711, −10.47055338524829, −9.956213342348396, −9.398088103826269, −8.645992578740621, −8.006822376515734, −7.797845730699759, −7.171763383575204, −6.292581523620245, −5.632853548086275, −5.041681483191735, −4.568120848853515, −3.911416378726030, −2.975241510199338, −2.265295203450861, −1.412398138835899, −0.8493046620120516,
0.8493046620120516, 1.412398138835899, 2.265295203450861, 2.975241510199338, 3.911416378726030, 4.568120848853515, 5.041681483191735, 5.632853548086275, 6.292581523620245, 7.171763383575204, 7.797845730699759, 8.006822376515734, 8.645992578740621, 9.398088103826269, 9.956213342348396, 10.47055338524829, 11.36524100348711, 11.56846370333840, 11.85450981447881, 12.67268660871602, 13.51047356691001, 13.82367783706395, 14.29789066608751, 14.82275895666786, 15.40398044100071
