| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s − 4·17-s + 4·19-s − 2·21-s − 4·23-s + 27-s − 6·29-s − 33-s − 2·37-s + 2·41-s − 2·43-s − 12·47-s − 3·49-s − 4·51-s − 2·53-s + 4·57-s − 4·59-s − 2·61-s − 2·63-s − 4·69-s + 8·71-s + 2·77-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.970·17-s + 0.917·19-s − 0.436·21-s − 0.834·23-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.328·37-s + 0.312·41-s − 0.304·43-s − 1.75·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.251·63-s − 0.481·69-s + 0.949·71-s + 0.227·77-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372518282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372518282\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.40807293786218, −13.91909140261372, −13.37290756536225, −13.09353479733853, −12.49096926632622, −12.02834988766317, −11.18592691280236, −11.05658960440674, −10.06268846970146, −9.828554828343956, −9.325219291716427, −8.754621101114414, −8.209329955821609, −7.579802102001380, −7.211248904970166, −6.353242294473667, −6.177653972118070, −5.190379497255566, −4.795472502321966, −3.896842050980002, −3.483546269410918, −2.868919060029919, −2.141409890635418, −1.536650176529709, −0.3764780051341747,
0.3764780051341747, 1.536650176529709, 2.141409890635418, 2.868919060029919, 3.483546269410918, 3.896842050980002, 4.795472502321966, 5.190379497255566, 6.177653972118070, 6.353242294473667, 7.211248904970166, 7.579802102001380, 8.209329955821609, 8.754621101114414, 9.325219291716427, 9.828554828343956, 10.06268846970146, 11.05658960440674, 11.18592691280236, 12.02834988766317, 12.49096926632622, 13.09353479733853, 13.37290756536225, 13.91909140261372, 14.40807293786218
