| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 6·11-s + 14-s + 16-s − 6·17-s − 2·19-s − 20-s + 6·22-s + 25-s − 28-s + 4·31-s − 32-s + 6·34-s + 35-s − 2·37-s + 2·38-s + 40-s − 6·41-s + 8·43-s − 6·44-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.80·11-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.223·20-s + 1.27·22-s + 1/5·25-s − 0.188·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s − 0.328·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s + 1.21·43-s − 0.904·44-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2794237633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2794237633\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 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.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 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.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.46987784521856, −13.23789945456564, −12.70677444845854, −12.25101573123831, −11.57226791441325, −11.20803782935162, −10.58058745767541, −10.35863216983300, −9.869592629419077, −9.129411290930604, −8.701283221570758, −8.269219504536140, −7.785097699330260, −7.259975536303295, −6.723444062385878, −6.311832718687191, −5.487318764534800, −5.123065287991402, −4.370268438164438, −3.847115733779026, −3.037558826805679, −2.432371399635040, −2.177642459791759, −1.043448716244661, −0.2039361447070195,
0.2039361447070195, 1.043448716244661, 2.177642459791759, 2.432371399635040, 3.037558826805679, 3.847115733779026, 4.370268438164438, 5.123065287991402, 5.487318764534800, 6.311832718687191, 6.723444062385878, 7.259975536303295, 7.785097699330260, 8.269219504536140, 8.701283221570758, 9.129411290930604, 9.869592629419077, 10.35863216983300, 10.58058745767541, 11.20803782935162, 11.57226791441325, 12.25101573123831, 12.70677444845854, 13.23789945456564, 13.46987784521856
