| L(s) = 1 | − 3-s + 4·7-s + 9-s − 4·11-s + 13-s − 2·17-s − 4·21-s − 27-s + 10·29-s + 4·31-s + 4·33-s − 2·37-s − 39-s + 6·41-s − 12·43-s + 9·49-s + 2·51-s + 6·53-s − 12·59-s + 2·61-s + 4·63-s − 8·67-s − 2·73-s − 16·77-s + 8·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.872·21-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s + 0.937·41-s − 1.82·43-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.503·63-s − 0.977·67-s − 0.234·73-s − 1.82·77-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091373217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091373217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.19453736185529, −13.69142455389404, −13.41615707501339, −12.64439250346775, −12.19778181578740, −11.65698534177179, −11.26189537855176, −10.73362988184234, −10.30170601315193, −9.950454079702909, −8.923062025652828, −8.589829221967866, −7.933289692162359, −7.719036011564201, −6.918878979437006, −6.376741285668369, −5.744650047324306, −5.139686602314625, −4.658134220660058, −4.432488633762000, −3.381863403161944, −2.665708045739181, −2.005528061755049, −1.304497496843213, −0.5347580677740317,
0.5347580677740317, 1.304497496843213, 2.005528061755049, 2.665708045739181, 3.381863403161944, 4.432488633762000, 4.658134220660058, 5.139686602314625, 5.744650047324306, 6.376741285668369, 6.918878979437006, 7.719036011564201, 7.933289692162359, 8.589829221967866, 8.923062025652828, 9.950454079702909, 10.30170601315193, 10.73362988184234, 11.26189537855176, 11.65698534177179, 12.19778181578740, 12.64439250346775, 13.41615707501339, 13.69142455389404, 14.19453736185529
