| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2·7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 2·13-s + 2·14-s − 15-s + 16-s − 2·17-s + 18-s − 8·19-s + 20-s − 2·21-s − 22-s + 4·23-s − 24-s + 25-s + 2·26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.436·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 89 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.29587602067919, −14.94847074801481, −14.34181518127054, −13.85754681969772, −13.16982608051089, −12.72633075102006, −12.56256390389280, −11.51616117175156, −11.12483699372225, −10.85478256221997, −10.34641259802888, −9.424142027429918, −8.992292453462620, −8.179939066959332, −7.756545598672591, −6.857052362690234, −6.510276268588609, −5.858093885209107, −5.292526670630418, −4.783787863698521, −4.142228581601884, −3.565077786148069, −2.493240848237549, −1.971820441571538, −1.231156385894471, 0,
1.231156385894471, 1.971820441571538, 2.493240848237549, 3.565077786148069, 4.142228581601884, 4.783787863698521, 5.292526670630418, 5.858093885209107, 6.510276268588609, 6.857052362690234, 7.756545598672591, 8.179939066959332, 8.992292453462620, 9.424142027429918, 10.34641259802888, 10.85478256221997, 11.12483699372225, 11.51616117175156, 12.56256390389280, 12.72633075102006, 13.16982608051089, 13.85754681969772, 14.34181518127054, 14.94847074801481, 15.29587602067919
