| L(s) = 1 | + 3·3-s − 4·5-s − 5·7-s + 6·9-s − 3·11-s + 4·13-s − 12·15-s − 3·17-s − 15·21-s + 11·25-s + 9·27-s − 5·29-s + 5·31-s − 9·33-s + 20·35-s + 3·37-s + 12·39-s + 2·41-s − 4·43-s − 24·45-s − 2·47-s + 18·49-s − 9·51-s + 6·53-s + 12·55-s − 7·59-s − 15·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s − 1.88·7-s + 2·9-s − 0.904·11-s + 1.10·13-s − 3.09·15-s − 0.727·17-s − 3.27·21-s + 11/5·25-s + 1.73·27-s − 0.928·29-s + 0.898·31-s − 1.56·33-s + 3.38·35-s + 0.493·37-s + 1.92·39-s + 0.312·41-s − 0.609·43-s − 3.57·45-s − 0.291·47-s + 18/7·49-s − 1.26·51-s + 0.824·53-s + 1.61·55-s − 0.911·59-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.312622702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312622702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.97846730969857, −12.61257127470659, −12.14926623201934, −11.52186187625212, −10.95580674968241, −10.53089280917472, −10.01954493463485, −9.465905276108393, −9.044200790331453, −8.633969329127679, −8.268467801190856, −7.686098649482039, −7.449283701067513, −6.897910863996526, −6.377448808745514, −5.902365223603164, −4.878367108301848, −4.365762129118074, −3.808230418042993, −3.545886575543636, −3.074124157388082, −2.732751589450879, −2.062035684168654, −1.059435106541707, −0.3070605194217229,
0.3070605194217229, 1.059435106541707, 2.062035684168654, 2.732751589450879, 3.074124157388082, 3.545886575543636, 3.808230418042993, 4.365762129118074, 4.878367108301848, 5.902365223603164, 6.377448808745514, 6.897910863996526, 7.449283701067513, 7.686098649482039, 8.268467801190856, 8.633969329127679, 9.044200790331453, 9.465905276108393, 10.01954493463485, 10.53089280917472, 10.95580674968241, 11.52186187625212, 12.14926623201934, 12.61257127470659, 12.97846730969857
