| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 6·11-s − 4·13-s − 15-s − 6·17-s + 2·21-s + 25-s − 27-s + 8·31-s − 6·33-s − 2·35-s + 8·37-s + 4·39-s + 12·41-s + 2·43-s + 45-s − 3·49-s + 6·51-s − 6·53-s + 6·55-s + 12·59-s − 2·61-s − 2·63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.43·31-s − 1.04·33-s − 0.338·35-s + 1.31·37-s + 0.640·39-s + 1.87·41-s + 0.304·43-s + 0.149·45-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.809·55-s + 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.670398194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.670398194\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.55023247399586, −12.06992062644264, −11.70515068126073, −11.16134198616269, −10.91995180267112, −10.20033955473419, −9.676105678981680, −9.410426187873679, −9.250893951512277, −8.428086167342097, −8.048845570702951, −7.190187081288552, −6.887107855283970, −6.547195927926309, −6.040387638848915, −5.784233927642569, −4.903017097544579, −4.433808857616885, −4.224197765331915, −3.503888547449088, −2.798904346021003, −2.313745447186618, −1.772201695851290, −0.8713083619748061, −0.5624118406021990,
0.5624118406021990, 0.8713083619748061, 1.772201695851290, 2.313745447186618, 2.798904346021003, 3.503888547449088, 4.224197765331915, 4.433808857616885, 4.903017097544579, 5.784233927642569, 6.040387638848915, 6.547195927926309, 6.887107855283970, 7.190187081288552, 8.048845570702951, 8.428086167342097, 9.250893951512277, 9.410426187873679, 9.676105678981680, 10.20033955473419, 10.91995180267112, 11.16134198616269, 11.70515068126073, 12.06992062644264, 12.55023247399586
