| L(s) = 1 | − 1.41i·2-s + (−0.5 − 2.59i)7-s − 2.82i·8-s − 1.41i·11-s − 5.19i·13-s + (−3.67 + 0.707i)14-s − 4.00·16-s − 7.34·17-s + 5.19i·19-s − 2.00·22-s + 7.07i·23-s − 7.34·26-s − 1.41i·29-s − 5.19i·31-s + 10.3i·34-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s + (−0.188 − 0.981i)7-s − 0.999i·8-s − 0.426i·11-s − 1.44i·13-s + (−0.981 + 0.188i)14-s − 1.00·16-s − 1.78·17-s + 1.19i·19-s − 0.426·22-s + 1.47i·23-s − 1.44·26-s − 0.262i·29-s − 0.933i·31-s + 1.78i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218079012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218079012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 2.59i)T \) |
| good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 5.19iT - 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 5.19iT - 97T^{2} \) |
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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
−9.349570675425657604867120640328, −8.073562081611305242529570400878, −7.51867194897739492065946659736, −6.50265606122229848944874024218, −5.76318395394248966154511831503, −4.40676383176610594170015999184, −3.65848891390443499347904285467, −2.85208741196689563831228447492, −1.65251161422378884792396157451, −0.43776668944705180736261646639,
2.05848124275456050316165314643, 2.67929590353215196777106880708, 4.48406787539151349840508467041, 4.88732293200713752147226885861, 6.21646216964931104442369312262, 6.56634674624358622498256895583, 7.21560368214974280975895388198, 8.364738522584268009643589649995, 8.944691042543722984464846165463, 9.423304098769723434972946637268
