| L(s) = 1 | + 3-s + 3·5-s + 9-s + 2·11-s + 13-s + 3·15-s − 2·17-s + 7·19-s − 23-s + 4·25-s + 27-s + 3·29-s + 31-s + 2·33-s + 8·37-s + 39-s + 2·41-s + 7·43-s + 3·45-s + 47-s − 2·51-s + 3·53-s + 6·55-s + 7·57-s + 12·59-s + 10·61-s + 3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.774·15-s − 0.485·17-s + 1.60·19-s − 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s + 0.179·31-s + 0.348·33-s + 1.31·37-s + 0.160·39-s + 0.312·41-s + 1.06·43-s + 0.447·45-s + 0.145·47-s − 0.280·51-s + 0.412·53-s + 0.809·55-s + 0.927·57-s + 1.56·59-s + 1.28·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.480630384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.480630384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.63179955927003, −13.18875975792343, −12.80433597614745, −12.02981681278436, −11.72439560383458, −11.06780558368629, −10.56013002257990, −9.931802020045658, −9.617210812889671, −9.282388019142546, −8.740025409151422, −8.229489124858835, −7.598951832949256, −7.054352077332948, −6.581736849740477, −5.921684655772024, −5.656448956815289, −4.965346210697997, −4.319528040739601, −3.781665198367355, −3.060733365753299, −2.502366344183896, −2.044997850897073, −1.223385170061679, −0.8267745081907896,
0.8267745081907896, 1.223385170061679, 2.044997850897073, 2.502366344183896, 3.060733365753299, 3.781665198367355, 4.319528040739601, 4.965346210697997, 5.656448956815289, 5.921684655772024, 6.581736849740477, 7.054352077332948, 7.598951832949256, 8.229489124858835, 8.740025409151422, 9.282388019142546, 9.617210812889671, 9.931802020045658, 10.56013002257990, 11.06780558368629, 11.72439560383458, 12.02981681278436, 12.80433597614745, 13.18875975792343, 13.63179955927003
