L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.22 − 2.34i)7-s − 0.999·8-s + (−2.03 − 1.17i)11-s + 4.64·13-s + (2.64 − 0.107i)14-s + (−0.5 − 0.866i)16-s + (−3.95 − 2.28i)17-s + (−0.491 + 0.283i)19-s − 2.35i·22-s + (−2.91 − 5.04i)23-s + (2.32 + 4.02i)26-s + (1.41 + 2.23i)28-s − 2.55i·29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.464 − 0.885i)7-s − 0.353·8-s + (−0.615 − 0.355i)11-s + 1.28·13-s + (0.706 − 0.0287i)14-s + (−0.125 − 0.216i)16-s + (−0.958 − 0.553i)17-s + (−0.112 + 0.0650i)19-s − 0.502i·22-s + (−0.607 − 1.05i)23-s + (0.455 + 0.789i)26-s + (0.267 + 0.422i)28-s − 0.474i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0156 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0156 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.156894782\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.156894782\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.22 + 2.34i)T \) |
good | 11 | \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + (3.95 + 2.28i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.491 - 0.283i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.91 + 5.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.55iT - 29T^{2} \) |
| 31 | \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.02 - 4.63i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 - 6.57iT - 43T^{2} \) |
| 47 | \( 1 + (5.46 - 3.15i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.07 + 10.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.67 + 2.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.85 - 3.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 2.00i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.02iT - 71T^{2} \) |
| 73 | \( 1 + (-4.10 + 7.11i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.13 + 7.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.171iT - 83T^{2} \) |
| 89 | \( 1 + (2.72 + 4.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.315176696565796243202814622380, −7.84771499461703230869693180226, −6.78411146389877017722614185450, −6.43666904042455568540333323342, −5.39818509631213904422033834000, −4.65383313823516150257516074289, −3.93346844291212078676549597930, −3.06562652647156814515968014621, −1.75840455364425734995649673574, −0.29806951271406331045874005760,
1.56579600883424547065173667266, 2.18626630238888384529277148676, 3.33972648364844027007901895654, 4.05996110117545032117562270388, 5.10689455519628062718134507076, 5.61296087763825294912199692706, 6.44027424956792225480947600337, 7.37497092806607697850525863408, 8.512429487444482529987242523702, 8.683673066068053302598271830673