| L(s) = 1 | − 5-s + 7-s − 3·11-s − 13-s − 17-s − 6·19-s + 6·23-s − 4·25-s − 2·29-s + 4·31-s − 35-s + 9·37-s − 7·43-s + 12·47-s + 49-s + 13·53-s + 3·55-s + 4·59-s + 2·61-s + 65-s − 13·67-s + 8·71-s + 73-s − 3·77-s − 5·79-s − 83-s + 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.904·11-s − 0.277·13-s − 0.242·17-s − 1.37·19-s + 1.25·23-s − 4/5·25-s − 0.371·29-s + 0.718·31-s − 0.169·35-s + 1.47·37-s − 1.06·43-s + 1.75·47-s + 1/7·49-s + 1.78·53-s + 0.404·55-s + 0.520·59-s + 0.256·61-s + 0.124·65-s − 1.58·67-s + 0.949·71-s + 0.117·73-s − 0.341·77-s − 0.562·79-s − 0.109·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−16.11274258880219, −15.38836139655011, −15.13417871926059, −14.71773691105665, −13.88328795512117, −13.23660621692643, −12.99491113391035, −12.22455454379202, −11.66755964875086, −11.12978382853435, −10.56138704811107, −10.10780682178615, −9.310009660873496, −8.629317443847013, −8.225015626122406, −7.503840044929346, −7.093271591021644, −6.236998993294283, −5.618871818312028, −4.867945656471690, −4.336989591454321, −3.662158835228458, −2.626065215270258, −2.228065680136573, −1.011877244129381, 0,
1.011877244129381, 2.228065680136573, 2.626065215270258, 3.662158835228458, 4.336989591454321, 4.867945656471690, 5.618871818312028, 6.236998993294283, 7.093271591021644, 7.503840044929346, 8.225015626122406, 8.629317443847013, 9.310009660873496, 10.10780682178615, 10.56138704811107, 11.12978382853435, 11.66755964875086, 12.22455454379202, 12.99491113391035, 13.23660621692643, 13.88328795512117, 14.71773691105665, 15.13417871926059, 15.38836139655011, 16.11274258880219
