| L(s) = 1 | + 2-s + 4-s − 2.23·5-s + (−2.23 − 1.41i)7-s + 8-s − 2.23·10-s − 5.65i·11-s + 4.47·13-s + (−2.23 − 1.41i)14-s + 16-s − 3.16i·17-s − 3.16i·19-s − 2.23·20-s − 5.65i·22-s − 4·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.999·5-s + (−0.845 − 0.534i)7-s + 0.353·8-s − 0.707·10-s − 1.70i·11-s + 1.24·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s − 0.766i·17-s − 0.725i·19-s − 0.499·20-s − 1.20i·22-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12506 - 1.06852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12506 - 1.06852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
| good | 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 9.89iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.07iT - 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.77285359403148905694971150248, −9.542089809853215406048914735188, −8.448425870994716460067954645164, −7.73929964869810858096226532867, −6.55672153043421169953770646027, −6.01356655802899931922537557546, −4.61801633390633026986912014511, −3.60902593833576481775329754996, −3.04889338686205213081454071793, −0.68351850848864430962191165507,
1.92442929857318984324501031329, 3.46597770519678438586775217107, 4.05228616292438623610707696108, 5.25840706049287998141382002902, 6.34231617464860660251047249185, 7.09316568140731712323441094442, 8.085144425274389292411064674791, 8.999112744992472716470177730755, 10.14741179615023585932295677023, 10.83615632866309036858099705064
