| L(s) = 1 | − 3-s + 9-s − 4·11-s + 13-s + 6·17-s + 8·19-s − 8·23-s − 27-s − 2·29-s + 8·31-s + 4·33-s − 10·37-s − 39-s + 6·41-s − 4·43-s − 7·49-s − 6·51-s − 14·53-s − 8·57-s + 12·59-s + 10·61-s − 8·67-s + 8·69-s + 14·73-s − 4·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 1.83·19-s − 1.66·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s − 49-s − 0.840·51-s − 1.92·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s − 0.977·67-s + 0.963·69-s + 1.63·73-s − 0.450·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543995095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543995095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.09802955044329, −13.84715392228623, −13.30514946740455, −12.55778527219685, −12.31396538084870, −11.69715835460193, −11.37344821116505, −10.66098785239871, −10.08206929290175, −9.862247756850245, −9.395814939804243, −8.237195187574822, −8.147183304996328, −7.570797751402903, −6.987070436524886, −6.316721000774767, −5.634423004309747, −5.377266757161087, −4.858917850688850, −4.027907392246661, −3.349396669958027, −2.909326735978040, −1.967598499279262, −1.253575628280757, −0.4668446157451165,
0.4668446157451165, 1.253575628280757, 1.967598499279262, 2.909326735978040, 3.349396669958027, 4.027907392246661, 4.858917850688850, 5.377266757161087, 5.634423004309747, 6.316721000774767, 6.987070436524886, 7.570797751402903, 8.147183304996328, 8.237195187574822, 9.395814939804243, 9.862247756850245, 10.08206929290175, 10.66098785239871, 11.37344821116505, 11.69715835460193, 12.31396538084870, 12.55778527219685, 13.30514946740455, 13.84715392228623, 14.09802955044329
