| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s + 4·13-s − 16-s + 4·17-s + 4·19-s + 20-s − 8·23-s + 25-s − 4·26-s + 2·29-s + 2·31-s − 5·32-s − 4·34-s + 6·37-s − 4·38-s − 3·40-s − 6·41-s + 8·46-s − 12·47-s − 7·49-s − 50-s − 4·52-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 1.10·13-s − 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.784·26-s + 0.371·29-s + 0.359·31-s − 0.883·32-s − 0.685·34-s + 0.986·37-s − 0.648·38-s − 0.474·40-s − 0.937·41-s + 1.17·46-s − 1.75·47-s − 49-s − 0.141·50-s − 0.554·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9853457827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9853457827\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.524938276906474846841795068167, −8.142237019281881376697064078871, −7.56585813576836116271146411242, −6.54841296108664801340741209034, −5.68221115513478548651262709107, −4.81870639342071423223660557558, −3.92160420674737471960996206735, −3.25249076314348644933967911462, −1.70269532217990211821603827162, −0.71394829430623663651197723082,
0.71394829430623663651197723082, 1.70269532217990211821603827162, 3.25249076314348644933967911462, 3.92160420674737471960996206735, 4.81870639342071423223660557558, 5.68221115513478548651262709107, 6.54841296108664801340741209034, 7.56585813576836116271146411242, 8.142237019281881376697064078871, 8.524938276906474846841795068167
