L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.5 − 1.53i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−1.30 − 0.951i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (0.309 − 0.224i)9-s + 10-s + (3.30 + 0.224i)11-s − 1.61·12-s + (0.618 − 0.449i)13-s + (−0.309 − 0.951i)14-s + (0.5 − 1.53i)15-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.288 − 0.888i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (−0.534 − 0.388i)6-s + (0.116 − 0.359i)7-s + (−0.109 − 0.336i)8-s + (0.103 − 0.0748i)9-s + 0.316·10-s + (0.997 + 0.0676i)11-s − 0.467·12-s + (0.171 − 0.124i)13-s + (−0.0825 − 0.254i)14-s + (0.129 − 0.397i)15-s + (−0.202 − 0.146i)16-s + (−0.121 − 0.0881i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18254 - 1.79491i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18254 - 1.79491i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.30 - 0.224i)T \) |
good | 3 | \( 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2} \) |
| 13 | \( 1 + (-0.618 + 0.449i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.809 + 2.48i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 + (-1.23 + 3.80i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.61 - 2.62i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.14 - 3.52i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.97 - 6.06i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 + (3.47 + 10.6i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2 - 1.45i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.427 - 1.31i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.47 - 4.70i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + (-5.23 - 3.80i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.5 - 10.7i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.47 + 6.88i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.0729 - 0.0530i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 + (11.1 - 8.11i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.14385320701297954344442956655, −9.405250132371369006069010562452, −8.265147087890703267371056077997, −7.00049301573007638728868220832, −6.66389345787293434210768230083, −5.70285091538498267135874672427, −4.51966688842777777231854246349, −3.49503906684165804982781291450, −2.11124974885591595399890502008, −1.03324414962351971240185390136,
1.84364340299893270352310087809, 3.50710872506245741197475264460, 4.32341869038854605450179696358, 5.19562047942409531131931030638, 5.98205391920920461629917085968, 6.89252203225853931192566563304, 8.027325502275114945087588074203, 9.044432177788156487664542954730, 9.602578255529839268349884895186, 10.68916169025125010008904971460