| L(s) = 1 | + 5-s − 3·9-s + 11-s + 17-s + 4·19-s − 2·23-s + 25-s − 6·29-s − 3·31-s + 8·37-s + 7·43-s − 3·45-s + 6·47-s − 7·49-s + 55-s − 15·59-s + 2·67-s − 5·73-s − 4·79-s + 9·81-s − 11·83-s + 85-s − 9·89-s + 4·95-s + 4·97-s − 3·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 0.301·11-s + 0.242·17-s + 0.917·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s − 0.538·31-s + 1.31·37-s + 1.06·43-s − 0.447·45-s + 0.875·47-s − 49-s + 0.134·55-s − 1.95·59-s + 0.244·67-s − 0.585·73-s − 0.450·79-s + 81-s − 1.20·83-s + 0.108·85-s − 0.953·89-s + 0.410·95-s + 0.406·97-s − 0.301·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.74698202481292, −13.02362155930399, −12.71749779968372, −12.15653168307980, −11.56384333417381, −11.27097051230533, −10.83884213782820, −10.17552903411163, −9.683949901182040, −9.194869789573289, −8.955654654903067, −8.234974094636916, −7.662878614239186, −7.411286683986231, −6.623826223954425, −6.054744452919025, −5.683111822687740, −5.339526014828797, −4.520830030338904, −4.061563940647401, −3.245054876493523, −2.945191813781788, −2.201636092188808, −1.587537782814966, −0.8460880755767096, 0,
0.8460880755767096, 1.587537782814966, 2.201636092188808, 2.945191813781788, 3.245054876493523, 4.061563940647401, 4.520830030338904, 5.339526014828797, 5.683111822687740, 6.054744452919025, 6.623826223954425, 7.411286683986231, 7.662878614239186, 8.234974094636916, 8.955654654903067, 9.194869789573289, 9.683949901182040, 10.17552903411163, 10.83884213782820, 11.27097051230533, 11.56384333417381, 12.15653168307980, 12.71749779968372, 13.02362155930399, 13.74698202481292
