| L(s) = 1 | + (0.498 + 1.32i)2-s + (−0.138 − 0.427i)3-s + (−1.50 + 1.31i)4-s + (−2.04 − 0.909i)5-s + (0.496 − 0.396i)6-s + (3.16 − 3.16i)7-s + (−2.49 − 1.33i)8-s + (2.26 − 1.64i)9-s + (0.185 − 3.15i)10-s + (−0.302 − 1.91i)11-s + (0.772 + 0.459i)12-s + (1.90 + 2.62i)13-s + (5.75 + 2.60i)14-s + (−0.104 + 0.999i)15-s + (0.519 − 3.96i)16-s + (−1.32 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.352 + 0.935i)2-s + (−0.0801 − 0.246i)3-s + (−0.751 + 0.659i)4-s + (−0.913 − 0.406i)5-s + (0.202 − 0.161i)6-s + (1.19 − 1.19i)7-s + (−0.882 − 0.471i)8-s + (0.754 − 0.548i)9-s + (0.0585 − 0.998i)10-s + (−0.0912 − 0.576i)11-s + (0.222 + 0.132i)12-s + (0.528 + 0.727i)13-s + (1.53 + 0.697i)14-s + (−0.0270 + 0.258i)15-s + (0.129 − 0.991i)16-s + (−0.322 − 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41074 - 0.0510762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41074 - 0.0510762i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.498 - 1.32i)T \) |
| 5 | \( 1 + (2.04 + 0.909i)T \) |
| good | 3 | \( 1 + (0.138 + 0.427i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-3.16 + 3.16i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.302 + 1.91i)T + (-10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 2.62i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.32 + 2.60i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.699 + 0.356i)T + (11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-0.505 - 3.19i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.0954 + 0.0486i)T + (17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (-7.25 - 2.35i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.32 + 5.94i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.87 + 8.08i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 3.10iT - 43T^{2} \) |
| 47 | \( 1 + (-2.89 + 5.67i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.573 + 1.76i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.12 - 13.4i)T + (-56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 8.56i)T + (-58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-5.57 - 1.81i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.86 - 11.8i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (16.4 - 2.60i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.79 - 11.6i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.03 + 9.32i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.78 - 1.29i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.86 + 0.950i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.57649274494766042089201004849, −10.42859130292897834635104041783, −9.005673806391509952917594034518, −8.307463956767665331037397276991, −7.27445319118065830725845278090, −6.95464879763059450497420319946, −5.36179129299653587374289164260, −4.32375498032527008959729927316, −3.75446807160909272229999860092, −0.966242500723580562381118827498,
1.74930672481255503649879709754, 3.03964619351170382109941216730, 4.45485681795499866809166578235, 4.97439841825090714814489963349, 6.32495571355800006532301286698, 7.991611101083060230082941118804, 8.415357527935740476736336015436, 9.757122124657256933625367086825, 10.68545189992987931970676186629, 11.23923963737358245568176254672
