| L(s) = 1 | + 1.19i·3-s − 1.19i·7-s + 1.56·9-s − 5.86·11-s − 0.364i·13-s − 1.19i·17-s + 19-s + 1.43·21-s − 8.23i·23-s + 5.46i·27-s + 7.86·29-s + 7.30·31-s − 7.03i·33-s − 7.13i·37-s + 0.436·39-s + ⋯ |
| L(s) = 1 | + 0.692i·3-s − 0.453i·7-s + 0.521·9-s − 1.76·11-s − 0.101i·13-s − 0.290i·17-s + 0.229·19-s + 0.313·21-s − 1.71i·23-s + 1.05i·27-s + 1.46·29-s + 1.31·31-s − 1.22i·33-s − 1.17i·37-s + 0.0699·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511586915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511586915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.19iT - 3T^{2} \) |
| 7 | \( 1 + 1.19iT - 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 + 0.364iT - 13T^{2} \) |
| 17 | \( 1 + 1.19iT - 17T^{2} \) |
| 23 | \( 1 + 8.23iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 + 7.13iT - 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 + 7.39iT - 43T^{2} \) |
| 47 | \( 1 - 13.7iT - 47T^{2} \) |
| 53 | \( 1 + 7.39iT - 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 - 2.12T + 71T^{2} \) |
| 73 | \( 1 - 2.50iT - 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 + 3.02iT - 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 + 1.09iT - 97T^{2} \) |
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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
−9.274911049862706154405392385941, −8.312040730119359439292507347042, −7.68429974285222849501183845670, −6.85185378325328302268672668287, −5.87630952079067468702004219079, −4.75677383604110831092637037227, −4.52683454106616675677045127272, −3.21856652070504497290577228947, −2.37359209906776269107050269907, −0.63183222987688180841715594808,
1.13317882476894614782603636894, 2.32098954266835548243413654495, 3.13169224929395554236910406435, 4.50002116371458927227463734861, 5.28277810624601498460993332308, 6.13433610791757439883841768957, 6.97363755156023619979738768779, 7.86058123014955986971715451766, 8.147391433163440784609323255897, 9.275654801003918793214927455766
