| L(s) = 1 | − 0.339i·3-s − 3.88·7-s + 2.88·9-s − 1.54i·11-s + (0.660 + 3.54i)13-s + 2.86i·19-s + 1.32i·21-s + 5.42i·23-s − 2i·27-s + 5.20·29-s − 6.22i·31-s − 0.524·33-s + 8.56·37-s + (1.20 − 0.224i)39-s + 9.08i·41-s + ⋯ |
| L(s) = 1 | − 0.196i·3-s − 1.46·7-s + 0.961·9-s − 0.465i·11-s + (0.183 + 0.983i)13-s + 0.657i·19-s + 0.288i·21-s + 1.13i·23-s − 0.384i·27-s + 0.966·29-s − 1.11i·31-s − 0.0913·33-s + 1.40·37-s + (0.192 − 0.0359i)39-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.390212925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390212925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.660 - 3.54i)T \) |
| good | 3 | \( 1 + 0.339iT - 3T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 1.54iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2.86iT - 19T^{2} \) |
| 23 | \( 1 - 5.42iT - 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 - 9.08iT - 41T^{2} \) |
| 43 | \( 1 - 0.980iT - 43T^{2} \) |
| 47 | \( 1 - 6.52T + 47T^{2} \) |
| 53 | \( 1 - 6.44iT - 53T^{2} \) |
| 59 | \( 1 - 4.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 6.97T + 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.43T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 + 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−9.678461228777771260241607906284, −9.173686086601090390979556516422, −8.019217485004238027106796894885, −7.19967714125383673023311097023, −6.40202639855208999461406355263, −5.85975294890561849708935494772, −4.40524909395197025641691726582, −3.68577637448596107085031520788, −2.58337627333393906930395201908, −1.14539724080367074714549133205,
0.69053805646061346127456495378, 2.46613558391985161865116519353, 3.41350393607633713325775023121, 4.35994846732414704451639407723, 5.33171589161812881838131260503, 6.48974188716872882602688234591, 6.89879824923318715405744128211, 7.915392574001518754132302314819, 8.938628379663681381447105959095, 9.640643301584127819034603664328
