| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 11-s + 2·13-s + 2·15-s − 17-s − 4·19-s − 4·21-s − 25-s − 27-s − 2·29-s − 33-s − 8·35-s + 2·37-s − 2·39-s + 6·41-s + 4·43-s − 2·45-s + 12·47-s + 9·49-s + 51-s + 2·53-s − 2·55-s + 4·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.242·17-s − 0.917·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.174·33-s − 1.35·35-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 9/7·49-s + 0.140·51-s + 0.274·53-s − 0.269·55-s + 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675399256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675399256\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.92507511265401, −14.56369290742748, −13.84399188420600, −13.43653685963825, −12.62075604654866, −12.14948800892176, −11.76360200121269, −11.13711364013715, −10.85679013114696, −10.51202165704931, −9.473790536916686, −8.962790594125844, −8.355011072291898, −7.873915307605309, −7.418559094614139, −6.818806904461179, −5.934663283780814, −5.666928471660559, −4.727249595421731, −4.228618289735131, −4.027911749364543, −2.938981317724842, −2.027389784323417, −1.380826441158703, −0.5288883880982531,
0.5288883880982531, 1.380826441158703, 2.027389784323417, 2.938981317724842, 4.027911749364543, 4.228618289735131, 4.727249595421731, 5.666928471660559, 5.934663283780814, 6.818806904461179, 7.418559094614139, 7.873915307605309, 8.355011072291898, 8.962790594125844, 9.473790536916686, 10.51202165704931, 10.85679013114696, 11.13711364013715, 11.76360200121269, 12.14948800892176, 12.62075604654866, 13.43653685963825, 13.84399188420600, 14.56369290742748, 14.92507511265401
