| L(s) = 1 | + (0.104 + 0.994i)2-s + (0.856 + 0.951i)3-s + (−0.978 + 0.207i)4-s + (−1.58 − 1.57i)5-s + (−0.856 + 0.951i)6-s + 1.37·7-s + (−0.309 − 0.951i)8-s + (0.142 − 1.35i)9-s + (1.39 − 1.74i)10-s + (−1.54 + 1.12i)11-s + (−1.03 − 0.752i)12-s + (0.223 − 2.13i)13-s + (0.144 + 1.37i)14-s + (0.134 − 2.85i)15-s + (0.913 − 0.406i)16-s + (5.26 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (0.494 + 0.549i)3-s + (−0.489 + 0.103i)4-s + (−0.710 − 0.703i)5-s + (−0.349 + 0.388i)6-s + 0.520·7-s + (−0.109 − 0.336i)8-s + (0.0474 − 0.451i)9-s + (0.442 − 0.551i)10-s + (−0.466 + 0.338i)11-s + (−0.298 − 0.217i)12-s + (0.0621 − 0.591i)13-s + (0.0384 + 0.366i)14-s + (0.0348 − 0.738i)15-s + (0.228 − 0.101i)16-s + (1.27 + 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66176 + 0.622570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66176 + 0.622570i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (1.58 + 1.57i)T \) |
| 19 | \( 1 + (-4.30 - 0.675i)T \) |
| good | 3 | \( 1 + (-0.856 - 0.951i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + (1.54 - 1.12i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.223 + 2.13i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-5.26 - 1.11i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (-3.10 - 1.38i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-5.04 + 1.07i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-1.90 - 5.85i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.36 + 6.08i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-5.91 + 2.63i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.828 + 1.43i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.75 - 0.585i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-13.6 + 2.90i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-9.43 + 4.20i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (12.6 + 5.64i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (3.72 - 4.13i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-0.658 - 0.731i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (0.399 + 3.79i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-4.86 - 5.40i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.02 - 6.24i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.17 - 2.75i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (0.371 + 0.412i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−9.927540323711475021134313799407, −9.127431813514904860534035958918, −8.354718734730975896645181879480, −7.79224016253415879085965679431, −6.95445952639768531826247011233, −5.50002933547480443830150261703, −4.99829903754199718639581195835, −3.90824963887383883905351939065, −3.15326565448379684912092072038, −1.01118572795356922307992494771,
1.18621194021028569291743742171, 2.58902681912140177731777475632, 3.23200097288191687438082552595, 4.47972271863431732577178363261, 5.40338406272634710998076576948, 6.77080965664913129358428409688, 7.67298280626792875430021672495, 8.105751928142015665831881520412, 9.067768481596868377103709684314, 10.21561327362764951310903772280
