| L(s) = 1 | + 3-s + 9-s − 4·11-s + 3·17-s + 6·19-s − 23-s + 27-s + 3·31-s − 4·33-s − 2·37-s − 41-s − 10·43-s + 47-s + 3·51-s − 6·53-s + 6·57-s − 6·59-s − 8·61-s + 8·67-s − 69-s − 71-s − 2·73-s − 79-s + 81-s − 12·83-s + 9·89-s + 3·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.727·17-s + 1.37·19-s − 0.208·23-s + 0.192·27-s + 0.538·31-s − 0.696·33-s − 0.328·37-s − 0.156·41-s − 1.52·43-s + 0.145·47-s + 0.420·51-s − 0.824·53-s + 0.794·57-s − 0.781·59-s − 1.02·61-s + 0.977·67-s − 0.120·69-s − 0.118·71-s − 0.234·73-s − 0.112·79-s + 1/9·81-s − 1.31·83-s + 0.953·89-s + 0.311·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.273348894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273348894\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.10527698634828, −12.39184907467934, −12.05166407045714, −11.61461212976603, −10.97877173308927, −10.50350352348837, −10.07342911536943, −9.647537135540365, −9.295708367577341, −8.527126083172166, −8.233793635277466, −7.626276938481582, −7.506755516589418, −6.777335236944551, −6.247954554014493, −5.595096651203770, −5.130868364121014, −4.807090229410046, −4.028300016074482, −3.428030693503339, −2.969459006790466, −2.616342574037603, −1.738565045383691, −1.294451323186590, −0.3962187358128352,
0.3962187358128352, 1.294451323186590, 1.738565045383691, 2.616342574037603, 2.969459006790466, 3.428030693503339, 4.028300016074482, 4.807090229410046, 5.130868364121014, 5.595096651203770, 6.247954554014493, 6.777335236944551, 7.506755516589418, 7.626276938481582, 8.233793635277466, 8.527126083172166, 9.295708367577341, 9.647537135540365, 10.07342911536943, 10.50350352348837, 10.97877173308927, 11.61461212976603, 12.05166407045714, 12.39184907467934, 13.10527698634828
