| L(s) = 1 | − 2·5-s − 3·11-s − 13-s + 3·17-s + 19-s − 4·23-s − 25-s − 3·29-s − 4·31-s + 2·37-s + 10·41-s + 6·43-s + 13·47-s − 9·53-s + 6·55-s − 11·59-s − 61-s + 2·65-s − 3·67-s + 13·71-s + 8·73-s − 14·79-s + 16·83-s − 6·85-s − 6·89-s − 2·95-s − 8·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s − 0.557·29-s − 0.718·31-s + 0.328·37-s + 1.56·41-s + 0.914·43-s + 1.89·47-s − 1.23·53-s + 0.809·55-s − 1.43·59-s − 0.128·61-s + 0.248·65-s − 0.366·67-s + 1.54·71-s + 0.936·73-s − 1.57·79-s + 1.75·83-s − 0.650·85-s − 0.635·89-s − 0.205·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9411563025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9411563025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.96174949600257, −13.37375166614757, −12.56710751497410, −12.49308574022143, −11.99963765773279, −11.31982043311514, −10.83850253837757, −10.61863579394606, −9.723876218741095, −9.478016003501231, −8.876818860968486, −8.020094505835165, −7.809282201488805, −7.516482434438627, −6.869513844737244, −6.005925129322067, −5.671839399731101, −5.106179938952274, −4.308581889714937, −3.999166534615188, −3.316581955247408, −2.671159697526390, −2.099575103006815, −1.165711446926433, −0.3310123142186899,
0.3310123142186899, 1.165711446926433, 2.099575103006815, 2.671159697526390, 3.316581955247408, 3.999166534615188, 4.308581889714937, 5.106179938952274, 5.671839399731101, 6.005925129322067, 6.869513844737244, 7.516482434438627, 7.809282201488805, 8.020094505835165, 8.876818860968486, 9.478016003501231, 9.723876218741095, 10.61863579394606, 10.83850253837757, 11.31982043311514, 11.99963765773279, 12.49308574022143, 12.56710751497410, 13.37375166614757, 13.96174949600257
