| L(s) = 1 | − 0.848i·3-s + 1.74i·7-s + 2.28·9-s + 5.92·11-s − 6.78i·13-s + 1.86i·17-s + 19-s + 1.48·21-s − 5.94i·23-s − 4.47i·27-s − 3.29·29-s − 5.75·31-s − 5.02i·33-s − 4.36i·37-s − 5.75·39-s + ⋯ |
| L(s) = 1 | − 0.489i·3-s + 0.659i·7-s + 0.760·9-s + 1.78·11-s − 1.88i·13-s + 0.451i·17-s + 0.229·19-s + 0.322·21-s − 1.23i·23-s − 0.862i·27-s − 0.612·29-s − 1.03·31-s − 0.874i·33-s − 0.717i·37-s − 0.921·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.270915742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.270915742\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.848iT - 3T^{2} \) |
| 7 | \( 1 - 1.74iT - 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 6.78iT - 13T^{2} \) |
| 17 | \( 1 - 1.86iT - 17T^{2} \) |
| 23 | \( 1 + 5.94iT - 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 + 4.36iT - 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 6.98iT - 43T^{2} \) |
| 47 | \( 1 + 4.02iT - 47T^{2} \) |
| 53 | \( 1 - 9.19iT - 53T^{2} \) |
| 59 | \( 1 + 2.51T + 59T^{2} \) |
| 61 | \( 1 + 2.49T + 61T^{2} \) |
| 67 | \( 1 - 6.90iT - 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 4.94iT - 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
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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
−8.324655027614379535576030737199, −7.60869879361256874068967772749, −6.93046094990278175217397234404, −6.08716290866800015680594243099, −5.63440434947250956652456405853, −4.46613191150594795364263193140, −3.74390137480486905519075226980, −2.74812043531637632592192925961, −1.69903707499535586907531091451, −0.75878320751210343692317020396,
1.22105117527398968286102999411, 1.91303258702036406006892527806, 3.60018044465088418457920987096, 3.96796751526747506118423040006, 4.58887128385372017041781616548, 5.60095463038775083830483832420, 6.80304683040596534375394544574, 6.87712206170750253262522843812, 7.74511386516323201435774703777, 9.014812229529090606949252830806
