| L(s) = 1 | + (1.63 + 0.577i)3-s + (−2.15 − 0.603i)5-s − 1.63·7-s + (2.33 + 1.88i)9-s − 0.207·11-s − 1.70i·13-s + (−3.16 − 2.22i)15-s + 5.57i·17-s + 2.31i·19-s + (−2.67 − 0.945i)21-s + (2.87 + 3.83i)23-s + (4.27 + 2.60i)25-s + (2.72 + 4.42i)27-s + 4.70i·29-s − 2.76·31-s + ⋯ |
| L(s) = 1 | + (0.942 + 0.333i)3-s + (−0.962 − 0.270i)5-s − 0.618·7-s + (0.777 + 0.628i)9-s − 0.0624·11-s − 0.472i·13-s + (−0.817 − 0.575i)15-s + 1.35i·17-s + 0.532i·19-s + (−0.583 − 0.206i)21-s + (0.600 + 0.799i)23-s + (0.854 + 0.520i)25-s + (0.523 + 0.851i)27-s + 0.873i·29-s − 0.496·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0303 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0303 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492782190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492782190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.63 - 0.577i)T \) |
| 5 | \( 1 + (2.15 + 0.603i)T \) |
| 23 | \( 1 + (-2.87 - 3.83i)T \) |
| good | 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 + 0.207T + 11T^{2} \) |
| 13 | \( 1 + 1.70iT - 13T^{2} \) |
| 17 | \( 1 - 5.57iT - 17T^{2} \) |
| 19 | \( 1 - 2.31iT - 19T^{2} \) |
| 29 | \( 1 - 4.70iT - 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 7.81iT - 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 + 4.72iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 8.69iT - 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 + 0.717iT - 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 + 4.32iT - 79T^{2} \) |
| 83 | \( 1 - 1.66iT - 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 9.51T + 97T^{2} \) |
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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
−9.681507118509136829972641996390, −8.897105198416046051010889202386, −8.186368818048154576960237668102, −7.61720855611081455848791550452, −6.70577488776861670683347349511, −5.51962347279757043030817635065, −4.46316048306501924417978272375, −3.61084948171327413960354788163, −3.03527872366744202383855699861, −1.47926350091498949236499440958,
0.55847508279453259286255999229, 2.36513872396822482257775094657, 3.15256852358693630077465985626, 4.02687546682059373307153296934, 4.94837662366319676806988950180, 6.49828415633536961902688146299, 7.01535670754160573331107500547, 7.72648014633381848752572522815, 8.556947211206269219007431509937, 9.294332682586057158608095054860
