| L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.217 + 0.670i)3-s + (0.309 − 0.951i)4-s + (0.980 − 2.00i)5-s + (−0.217 − 0.670i)6-s − 4.30·7-s + (0.309 + 0.951i)8-s + (2.02 + 1.47i)9-s + (0.388 + 2.20i)10-s + (−1.42 + 1.03i)11-s + (0.570 + 0.414i)12-s + (−0.809 − 0.587i)13-s + (3.48 − 2.52i)14-s + (1.13 + 1.09i)15-s + (−0.809 − 0.587i)16-s + (−1.12 − 3.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.125 + 0.386i)3-s + (0.154 − 0.475i)4-s + (0.438 − 0.898i)5-s + (−0.0889 − 0.273i)6-s − 1.62·7-s + (0.109 + 0.336i)8-s + (0.675 + 0.490i)9-s + (0.122 + 0.696i)10-s + (−0.429 + 0.311i)11-s + (0.164 + 0.119i)12-s + (−0.224 − 0.163i)13-s + (0.930 − 0.676i)14-s + (0.292 + 0.282i)15-s + (−0.202 − 0.146i)16-s + (−0.272 − 0.839i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00702099 - 0.0290320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00702099 - 0.0290320i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.980 + 2.00i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| good | 3 | \( 1 + (0.217 - 0.670i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 11 | \( 1 + (1.42 - 1.03i)T + (3.39 - 10.4i)T^{2} \) |
| 17 | \( 1 + (1.12 + 3.46i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.417 + 1.28i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.83 - 3.51i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.16 - 9.75i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.436 - 1.34i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (7.27 + 5.28i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (8.06 + 5.86i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 3.13T + 43T^{2} \) |
| 47 | \( 1 + (0.687 - 2.11i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.30 + 10.1i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (4.95 + 3.59i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.86 - 2.08i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.94 - 9.06i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.15 + 9.69i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.34 - 2.42i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.29 - 10.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.06 + 12.4i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-10.7 + 7.82i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.46 - 4.51i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.01401319110864258308630177711, −9.391030822056274227360646141832, −8.707844576709756346002429664833, −7.42058706436992050613656290925, −6.77174637518554977235698608792, −5.56429650601539529846393938990, −4.92573149027623943887022869472, −3.53073661702004839711992711689, −1.93857881230682497550995519947, −0.01821434278659446744298705056,
1.97662926667095379142530941574, 3.10342611120052893623085554533, 4.01192158277553501581968185703, 6.07816243350558333065267019627, 6.42971344394766075990159620291, 7.30818780692083884391908054613, 8.347987383694997133192597886183, 9.596732784189283763106423379178, 9.988812489336514864666493455523, 10.60245050891481525091037243110
