| L(s) = 1 | − 0.517i·2-s + 1.73·4-s + (−1 + 2.44i)7-s − 1.93i·8-s − 0.378i·11-s + 1.79i·13-s + (1.26 + 0.517i)14-s + 2.46·16-s + 3.46·17-s + 1.79i·19-s − 0.196·22-s − 1.41i·23-s + 0.928·26-s + (−1.73 + 4.24i)28-s + 1.41i·29-s + ⋯ |
| L(s) = 1 | − 0.366i·2-s + 0.866·4-s + (−0.377 + 0.925i)7-s − 0.683i·8-s − 0.114i·11-s + 0.497i·13-s + (0.338 + 0.138i)14-s + 0.616·16-s + 0.840·17-s + 0.411i·19-s − 0.0418·22-s − 0.294i·23-s + 0.182·26-s + (−0.327 + 0.801i)28-s + 0.262i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103886359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103886359\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| good | 2 | \( 1 + 0.517iT - 2T^{2} \) |
| 11 | \( 1 + 0.378iT - 11T^{2} \) |
| 13 | \( 1 - 1.79iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 1.79iT - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.69iT - 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 13.3iT - 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 15.0iT - 71T^{2} \) |
| 73 | \( 1 + 1.79iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−9.461096036137354920604949430926, −8.823349683230176401046452223194, −7.77425408754390105719865895211, −7.05387617105105259549537302757, −6.09232463331881387388497538460, −5.62233514719012109127896496325, −4.27069684410793498293823010094, −3.16782251333163720374521242951, −2.46735124269486973015056308051, −1.29133477430316906681036468165,
0.902393286225419105263988306086, 2.35121658388123257591330787944, 3.34008550120200504771844289341, 4.32850318315833994956933670068, 5.55724036889274050239037891694, 6.15700566104011414588971634033, 7.16540904195218840067016307857, 7.56478860391856785871368554333, 8.344723529421485985295023266069, 9.591456195971934149379275842215
