| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 5·11-s + 12-s − 15-s + 16-s − 2·17-s + 18-s − 2·19-s − 20-s + 5·22-s + 23-s + 24-s + 25-s + 27-s − 29-s − 30-s + 31-s + 32-s + 5·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 1.06·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.185·29-s − 0.182·30-s + 0.179·31-s + 0.176·32-s + 0.870·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.031077494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.031077494\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.86444789114202, −12.51568553828496, −11.97019579838540, −11.46496145317064, −11.24898036855194, −10.60519107475251, −10.16261462083003, −9.474379872118831, −8.991567570889541, −8.853052973391537, −8.072903958850992, −7.622550943325739, −7.157661864826226, −6.705425416252266, −6.072649320856287, −5.913247649197452, −4.897542237928564, −4.555251326824643, −4.000941895000560, −3.718106515253138, −3.116923355376886, −2.396957091734401, −2.035624713150768, −1.170208619198947, −0.6629443438565942,
0.6629443438565942, 1.170208619198947, 2.035624713150768, 2.396957091734401, 3.116923355376886, 3.718106515253138, 4.000941895000560, 4.555251326824643, 4.897542237928564, 5.913247649197452, 6.072649320856287, 6.705425416252266, 7.157661864826226, 7.622550943325739, 8.072903958850992, 8.853052973391537, 8.991567570889541, 9.474379872118831, 10.16261462083003, 10.60519107475251, 11.24898036855194, 11.46496145317064, 11.97019579838540, 12.51568553828496, 12.86444789114202
