| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s − 3·11-s − 3·13-s + 3·14-s + 16-s − 4·19-s − 3·22-s + 5·23-s − 3·26-s + 3·28-s + 8·29-s + 8·31-s + 32-s + 2·37-s − 4·38-s − 6·41-s − 10·43-s − 3·44-s + 5·46-s − 3·47-s + 2·49-s − 3·52-s − 3·53-s + 3·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s − 0.904·11-s − 0.832·13-s + 0.801·14-s + 1/4·16-s − 0.917·19-s − 0.639·22-s + 1.04·23-s − 0.588·26-s + 0.566·28-s + 1.48·29-s + 1.43·31-s + 0.176·32-s + 0.328·37-s − 0.648·38-s − 0.937·41-s − 1.52·43-s − 0.452·44-s + 0.737·46-s − 0.437·47-s + 2/7·49-s − 0.416·52-s − 0.412·53-s + 0.400·56-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.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.69018621431659, −13.32067285702830, −12.76774489884156, −12.30084841760007, −11.85021001952092, −11.42723751192424, −10.81590185232958, −10.55064012659506, −9.930210731586325, −9.530377600033074, −8.522103757376685, −8.259586835545202, −7.998128621558603, −7.172802782201435, −6.774120235901330, −6.307431693941381, −5.502516447558305, −5.047607566284240, −4.629617887522080, −4.417934876542943, −3.380177118165872, −2.858011467511151, −2.371139271905286, −1.709210048767065, −1.009031371307510, 0,
1.009031371307510, 1.709210048767065, 2.371139271905286, 2.858011467511151, 3.380177118165872, 4.417934876542943, 4.629617887522080, 5.047607566284240, 5.502516447558305, 6.307431693941381, 6.774120235901330, 7.172802782201435, 7.998128621558603, 8.259586835545202, 8.522103757376685, 9.530377600033074, 9.930210731586325, 10.55064012659506, 10.81590185232958, 11.42723751192424, 11.85021001952092, 12.30084841760007, 12.76774489884156, 13.32067285702830, 13.69018621431659
