| L(s) = 1 | − 2-s + 4-s − 8-s − 11-s − 2·13-s + 16-s − 2·17-s + 8·19-s + 22-s + 4·23-s + 2·26-s − 2·29-s + 8·31-s − 32-s + 2·34-s + 2·37-s − 8·38-s − 6·41-s − 8·43-s − 44-s − 4·46-s − 4·47-s − 7·49-s − 2·52-s + 2·53-s + 2·58-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 1.83·19-s + 0.213·22-s + 0.834·23-s + 0.392·26-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 1.29·38-s − 0.937·41-s − 1.21·43-s − 0.150·44-s − 0.589·46-s − 0.583·47-s − 49-s − 0.277·52-s + 0.274·53-s + 0.262·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.249091255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249091255\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 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 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.195744637338893084617921509828, −7.64788230419804124569321376229, −6.91861228132228236468013108499, −6.31230116151273469639620393658, −5.22812246426631527724369203313, −4.79768983176545412480322545670, −3.45561642091890466591487384189, −2.82776324623515208830271662074, −1.77392224632628903464032451485, −0.68534793497909805647660080979,
0.68534793497909805647660080979, 1.77392224632628903464032451485, 2.82776324623515208830271662074, 3.45561642091890466591487384189, 4.79768983176545412480322545670, 5.22812246426631527724369203313, 6.31230116151273469639620393658, 6.91861228132228236468013108499, 7.64788230419804124569321376229, 8.195744637338893084617921509828
