| L(s) = 1 | − 3-s + 2·5-s − 3·7-s + 9-s − 11-s + 13-s − 2·15-s − 5·19-s + 3·21-s − 6·23-s − 25-s − 27-s + 8·29-s + 7·31-s + 33-s − 6·35-s + 37-s − 39-s + 5·43-s + 2·45-s + 4·47-s + 2·49-s + 12·53-s − 2·55-s + 5·57-s + 61-s − 3·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.516·15-s − 1.14·19-s + 0.654·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.25·31-s + 0.174·33-s − 1.01·35-s + 0.164·37-s − 0.160·39-s + 0.762·43-s + 0.298·45-s + 0.583·47-s + 2/7·49-s + 1.64·53-s − 0.269·55-s + 0.662·57-s + 0.128·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684683867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684683867\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−13.21307685086605, −12.92593219025158, −12.40338258791840, −11.93351285017199, −11.55556025534406, −10.70792784902242, −10.38041893480176, −10.03108504364267, −9.722050451774622, −8.994246248926836, −8.602610333513193, −7.995195464492958, −7.412717461836165, −6.667245722203187, −6.324384504296812, −6.084105148330069, −5.553415364457013, −4.891264099547732, −4.253835143190338, −3.843342338133745, −3.031877775409345, −2.408530616456116, −2.037758555646160, −1.051747413642080, −0.4401688554225551,
0.4401688554225551, 1.051747413642080, 2.037758555646160, 2.408530616456116, 3.031877775409345, 3.843342338133745, 4.253835143190338, 4.891264099547732, 5.553415364457013, 6.084105148330069, 6.324384504296812, 6.667245722203187, 7.412717461836165, 7.995195464492958, 8.602610333513193, 8.994246248926836, 9.722050451774622, 10.03108504364267, 10.38041893480176, 10.70792784902242, 11.55556025534406, 11.93351285017199, 12.40338258791840, 12.92593219025158, 13.21307685086605
