| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 5·11-s − 12-s + 13-s − 16-s + 7·17-s + 18-s + 19-s − 5·22-s − 8·23-s − 3·24-s + 26-s + 27-s − 6·29-s − 31-s + 5·32-s − 5·33-s + 7·34-s − 36-s + 10·37-s + 38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.277·13-s − 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.229·19-s − 1.06·22-s − 1.66·23-s − 0.612·24-s + 0.196·26-s + 0.192·27-s − 1.11·29-s − 0.179·31-s + 0.883·32-s − 0.870·33-s + 1.20·34-s − 1/6·36-s + 1.64·37-s + 0.162·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.180102002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180102002\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.44986341432843, −13.93346678157687, −13.70574670428360, −13.03042972917097, −12.74923947022969, −12.11109099337362, −11.77302940401320, −10.89243214589129, −10.31723657623504, −9.863535376369196, −9.437277043329379, −8.766989018432734, −8.114584880164213, −7.762180012997775, −7.388618592420434, −6.287327638677190, −5.743447128695097, −5.441387749185004, −4.746212537238711, −4.064921291805683, −3.568806228329295, −2.972007495252965, −2.405836322128252, −1.493595381867953, −0.4469956083550120,
0.4469956083550120, 1.493595381867953, 2.405836322128252, 2.972007495252965, 3.568806228329295, 4.064921291805683, 4.746212537238711, 5.441387749185004, 5.743447128695097, 6.287327638677190, 7.388618592420434, 7.762180012997775, 8.114584880164213, 8.766989018432734, 9.437277043329379, 9.863535376369196, 10.31723657623504, 10.89243214589129, 11.77302940401320, 12.11109099337362, 12.74923947022969, 13.03042972917097, 13.70574670428360, 13.93346678157687, 14.44986341432843
