| L(s) = 1 | + (0.891 − 0.453i)2-s + (−1.37 − 1.05i)3-s + (0.587 − 0.809i)4-s + (−1.04 − 1.97i)5-s + (−1.70 − 0.313i)6-s + (0.462 − 0.462i)7-s + (0.156 − 0.987i)8-s + (0.784 + 2.89i)9-s + (−1.82 − 1.28i)10-s + (−2.73 − 0.888i)11-s + (−1.66 + 0.494i)12-s + (1.97 − 3.86i)13-s + (0.202 − 0.621i)14-s + (−0.641 + 3.81i)15-s + (−0.309 − 0.951i)16-s + (5.75 + 0.911i)17-s + ⋯ |
| L(s) = 1 | + (0.630 − 0.321i)2-s + (−0.794 − 0.607i)3-s + (0.293 − 0.404i)4-s + (−0.467 − 0.883i)5-s + (−0.695 − 0.127i)6-s + (0.174 − 0.174i)7-s + (0.0553 − 0.349i)8-s + (0.261 + 0.965i)9-s + (−0.578 − 0.406i)10-s + (−0.824 − 0.267i)11-s + (−0.479 + 0.142i)12-s + (0.546 − 1.07i)13-s + (0.0539 − 0.166i)14-s + (−0.165 + 0.986i)15-s + (−0.0772 − 0.237i)16-s + (1.39 + 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.725142 - 0.899315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.725142 - 0.899315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 + 0.453i)T \) |
| 3 | \( 1 + (1.37 + 1.05i)T \) |
| 5 | \( 1 + (1.04 + 1.97i)T \) |
| good | 7 | \( 1 + (-0.462 + 0.462i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.73 + 0.888i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.97 + 3.86i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-5.75 - 0.911i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-4.28 - 5.89i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.316 - 0.622i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-2.60 - 1.89i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.82 - 3.50i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.93 + 0.988i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-6.34 + 2.06i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.08 + 5.08i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.474 + 2.99i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-2.23 + 0.353i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (2.66 + 8.19i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.77 + 8.55i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.22 - 14.0i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (7.15 - 9.84i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.498 + 0.254i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-6.00 + 8.25i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.742 - 4.68i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (1.78 - 5.49i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.26 + 0.833i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.56895933627508432661524366872, −12.06586124632489045949366392394, −10.92689824089302746884274231239, −10.09106821597752630992854368201, −8.217074764346506085161274625657, −7.48893963352225065078903368866, −5.68595363049078333295539595573, −5.24235495659356360620187352331, −3.52798553401659911512543196569, −1.19148255874697230776117067122,
3.10807683729975728615594109655, 4.44381569655266412161972951696, 5.57884745750058843778570604333, 6.74055420285992137469560405603, 7.70788312338096007230901557578, 9.374817315283107429471912084036, 10.54677827648746695885471970925, 11.46450600693531104496420334671, 12.01723051866921715171155560632, 13.38266210382368270420713169821
