| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 6·11-s + 12-s − 13-s + 16-s − 4·17-s − 18-s − 7·19-s + 6·22-s − 2·23-s − 24-s + 26-s + 27-s − 3·29-s + 2·31-s − 32-s − 6·33-s + 4·34-s + 36-s − 3·37-s + 7·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.60·19-s + 1.27·22-s − 0.417·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 1.04·33-s + 0.685·34-s + 1/6·36-s − 0.493·37-s + 1.13·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.06365446480780, −13.48123628057217, −12.99993941242207, −12.67679602411618, −12.22827222670557, −11.34336352671601, −10.95808422464668, −10.49129419501487, −10.21164964909821, −9.448094458357601, −9.127393865472752, −8.419953801733881, −8.153881953167488, −7.686086518877799, −7.082550923403910, −6.643979323933779, −5.875688085647415, −5.471662962259175, −4.609077301983093, −4.237050941802099, −3.462133514720235, −2.555103069501635, −2.403664102598472, −1.852113952489732, −0.6919192804850506, 0,
0.6919192804850506, 1.852113952489732, 2.403664102598472, 2.555103069501635, 3.462133514720235, 4.237050941802099, 4.609077301983093, 5.471662962259175, 5.875688085647415, 6.643979323933779, 7.082550923403910, 7.686086518877799, 8.153881953167488, 8.419953801733881, 9.127393865472752, 9.448094458357601, 10.21164964909821, 10.49129419501487, 10.95808422464668, 11.34336352671601, 12.22827222670557, 12.67679602411618, 12.99993941242207, 13.48123628057217, 14.06365446480780
