| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 13-s − 3·14-s + 16-s − 4·23-s − 26-s − 3·28-s − 4·29-s − 5·31-s + 32-s + 8·37-s − 4·43-s − 4·46-s − 8·47-s + 2·49-s − 52-s − 11·53-s − 3·56-s − 4·58-s − 5·62-s + 64-s + 8·67-s + 15·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.834·23-s − 0.196·26-s − 0.566·28-s − 0.742·29-s − 0.898·31-s + 0.176·32-s + 1.31·37-s − 0.609·43-s − 0.589·46-s − 1.16·47-s + 2/7·49-s − 0.138·52-s − 1.51·53-s − 0.400·56-s − 0.525·58-s − 0.635·62-s + 1/8·64-s + 0.977·67-s + 1.78·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.66846935621239, −13.09646707107380, −12.84294800677749, −12.47817777897604, −11.89676255467486, −11.26244021628570, −11.09629117236419, −10.23950705686353, −9.914975513384974, −9.421528920409081, −9.029172681327205, −8.145480807699950, −7.793717755906989, −7.253737183409520, −6.480833319748516, −6.395721056643515, −5.766202673945584, −5.148401298841325, −4.661894538481387, −3.972838784770349, −3.396285442605456, −3.161462666678121, −2.198956430285134, −1.897163411969470, −0.7907733138332456, 0,
0.7907733138332456, 1.897163411969470, 2.198956430285134, 3.161462666678121, 3.396285442605456, 3.972838784770349, 4.661894538481387, 5.148401298841325, 5.766202673945584, 6.395721056643515, 6.480833319748516, 7.253737183409520, 7.793717755906989, 8.145480807699950, 9.029172681327205, 9.421528920409081, 9.914975513384974, 10.23950705686353, 11.09629117236419, 11.26244021628570, 11.89676255467486, 12.47817777897604, 12.84294800677749, 13.09646707107380, 13.66846935621239
