| L(s) = 1 | + 11-s + 4·13-s − 17-s − 6·19-s + 4·23-s − 8·29-s + 6·31-s + 12·41-s − 4·43-s − 6·47-s − 7·49-s − 6·53-s + 2·61-s − 4·67-s − 8·71-s − 14·73-s − 8·79-s + 12·83-s + 8·89-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 1.10·13-s − 0.242·17-s − 1.37·19-s + 0.834·23-s − 1.48·29-s + 1.07·31-s + 1.87·41-s − 0.609·43-s − 0.875·47-s − 49-s − 0.824·53-s + 0.256·61-s − 0.488·67-s − 0.949·71-s − 1.63·73-s − 0.900·79-s + 1.31·83-s + 0.847·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.795871666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.795871666\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.64565251325623, −12.18089179351989, −11.50088321735307, −11.23990087030712, −10.81361289808525, −10.47064997433234, −9.757936038062570, −9.381675129931152, −8.903161932466269, −8.467453900510230, −8.097083360105399, −7.494851863767491, −6.996956874821164, −6.415367262472460, −6.129527573177074, −5.686404999778852, −4.957086707007810, −4.412961468695267, −4.121547084604448, −3.406245943710526, −3.006468325605052, −2.293138751436161, −1.666846646944975, −1.204306186098482, −0.3567919893761227,
0.3567919893761227, 1.204306186098482, 1.666846646944975, 2.293138751436161, 3.006468325605052, 3.406245943710526, 4.121547084604448, 4.412961468695267, 4.957086707007810, 5.686404999778852, 6.129527573177074, 6.415367262472460, 6.996956874821164, 7.494851863767491, 8.097083360105399, 8.467453900510230, 8.903161932466269, 9.381675129931152, 9.757936038062570, 10.47064997433234, 10.81361289808525, 11.23990087030712, 11.50088321735307, 12.18089179351989, 12.64565251325623
