| L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.604 − 0.128i)3-s + (−0.978 − 0.207i)4-s + (1.11 + 1.93i)5-s + (−0.190 + 0.587i)6-s + (−0.5 − 2.59i)7-s + (−0.309 + 0.951i)8-s + (−2.39 − 1.06i)9-s + (2.04 − 0.909i)10-s + (3.52 − 1.56i)11-s + (0.564 + 0.251i)12-s + (4.73 − 3.44i)13-s + (−2.63 + 0.225i)14-s + (−0.427 − 1.31i)15-s + (0.913 + 0.406i)16-s + (−5.15 − 5.72i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (−0.349 − 0.0741i)3-s + (−0.489 − 0.103i)4-s + (0.499 + 0.866i)5-s + (−0.0779 + 0.239i)6-s + (−0.188 − 0.981i)7-s + (−0.109 + 0.336i)8-s + (−0.797 − 0.354i)9-s + (0.645 − 0.287i)10-s + (1.06 − 0.472i)11-s + (0.162 + 0.0725i)12-s + (1.31 − 0.954i)13-s + (−0.704 + 0.0603i)14-s + (−0.110 − 0.339i)15-s + (0.228 + 0.101i)16-s + (−1.25 − 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.812045 - 0.900882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.812045 - 0.900882i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.11 - 1.93i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| good | 3 | \( 1 + (0.604 + 0.128i)T + (2.74 + 1.22i)T^{2} \) |
| 11 | \( 1 + (-3.52 + 1.56i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-4.73 + 3.44i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (5.15 + 5.72i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.86 + 1.24i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.153 - 1.46i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-1.07 - 3.30i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.24 + 3.60i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.348 - 0.155i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (7.59 - 5.51i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + (-6.18 + 6.86i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.230 - 0.0490i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.556 - 5.29i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (0.860 - 8.19i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (0.669 + 0.743i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (-4.92 - 15.1i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.91 + 2.18i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-5.98 + 6.64i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (1.78 - 5.48i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.617 + 5.87i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (3.47 + 10.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.30849066354010752360079351529, −10.58114994447734340322625948340, −9.546746367161768886989602224522, −8.745692508459953592660870546394, −7.23469469834820627015431364307, −6.36649768006400424686086358900, −5.38149083458543520037374250645, −3.73319296430241795703031561834, −2.96276881853183295611850008525, −0.945997287779039344000202784085,
1.80047837814745549072661691954, 3.88854438020718198155924047007, 5.04693684316809115115052535349, 6.02927800389679046034647376876, 6.51584301517084398969121143864, 8.267926848828567133256930611467, 8.887445021795441975078387280063, 9.455545139154867113280429976714, 10.91809148760484969689984634975, 11.89497108566264166629864588527
