| L(s) = 1 | + 5-s − 2·7-s + 6·13-s + 7·17-s − 7·19-s + 7·23-s + 25-s − 6·29-s + 3·31-s − 2·35-s + 6·37-s + 4·41-s − 8·43-s − 4·47-s − 3·49-s + 5·53-s − 6·59-s + 3·61-s + 6·65-s + 10·67-s + 12·71-s + 16·73-s + 79-s − 9·83-s + 7·85-s − 4·89-s − 12·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.66·13-s + 1.69·17-s − 1.60·19-s + 1.45·23-s + 1/5·25-s − 1.11·29-s + 0.538·31-s − 0.338·35-s + 0.986·37-s + 0.624·41-s − 1.21·43-s − 0.583·47-s − 3/7·49-s + 0.686·53-s − 0.781·59-s + 0.384·61-s + 0.744·65-s + 1.22·67-s + 1.42·71-s + 1.87·73-s + 0.112·79-s − 0.987·83-s + 0.759·85-s − 0.423·89-s − 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.356670231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356670231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−7.956292546282011086307331405765, −6.80515008937297405544034160598, −6.47430486987025233244155166175, −5.75763287819342312402253833932, −5.17026476610089217973521521370, −4.07015266642460281859770762634, −3.47839539026755024777673066155, −2.76291990114691511888286402155, −1.65358715019079043845736670439, −0.78452968369811149200725923167,
0.78452968369811149200725923167, 1.65358715019079043845736670439, 2.76291990114691511888286402155, 3.47839539026755024777673066155, 4.07015266642460281859770762634, 5.17026476610089217973521521370, 5.75763287819342312402253833932, 6.47430486987025233244155166175, 6.80515008937297405544034160598, 7.956292546282011086307331405765
