| L(s) = 1 | + 2·3-s + 4·5-s − 7-s + 9-s + 13-s + 8·15-s − 17-s − 2·21-s − 3·23-s + 11·25-s − 4·27-s − 6·31-s − 4·35-s − 3·37-s + 2·39-s − 2·41-s + 5·43-s + 4·45-s − 6·47-s − 6·49-s − 2·51-s + 6·53-s − 3·59-s + 14·61-s − 63-s + 4·65-s + 6·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.277·13-s + 2.06·15-s − 0.242·17-s − 0.436·21-s − 0.625·23-s + 11/5·25-s − 0.769·27-s − 1.07·31-s − 0.676·35-s − 0.493·37-s + 0.320·39-s − 0.312·41-s + 0.762·43-s + 0.596·45-s − 0.875·47-s − 6/7·49-s − 0.280·51-s + 0.824·53-s − 0.390·59-s + 1.79·61-s − 0.125·63-s + 0.496·65-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 628 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 628 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.643924654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.643924654\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 157 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−10.24493022670560792350810929195, −9.613176133662380360659095723198, −9.004575862383982599112358908454, −8.232107019684374222537788129761, −6.98996620066917499593591455989, −6.08593644475255391439556942950, −5.24926294111377476287433302961, −3.70669921189436073695957624365, −2.60774478575569127066437105876, −1.77480808282081288374631928121,
1.77480808282081288374631928121, 2.60774478575569127066437105876, 3.70669921189436073695957624365, 5.24926294111377476287433302961, 6.08593644475255391439556942950, 6.98996620066917499593591455989, 8.232107019684374222537788129761, 9.004575862383982599112358908454, 9.613176133662380360659095723198, 10.24493022670560792350810929195
