L(s) = 1 | + (−1.06 + 0.929i)2-s + (0.5 + 0.866i)3-s + (0.272 − 1.98i)4-s + (−2.12 − 1.22i)5-s + (−1.33 − 0.458i)6-s + (1.55 + 2.36i)8-s + (−0.499 + 0.866i)9-s + (3.40 − 0.667i)10-s + (−1.09 + 0.632i)11-s + (1.85 − 0.755i)12-s + 2.99i·13-s − 2.45i·15-s + (−3.85 − 1.07i)16-s + (−1.58 + 0.916i)17-s + (−0.272 − 1.38i)18-s + (2.07 − 3.60i)19-s + ⋯ |
L(s) = 1 | + (−0.753 + 0.657i)2-s + (0.288 + 0.499i)3-s + (0.136 − 0.990i)4-s + (−0.949 − 0.548i)5-s + (−0.546 − 0.187i)6-s + (0.548 + 0.836i)8-s + (−0.166 + 0.288i)9-s + (1.07 − 0.210i)10-s + (−0.330 + 0.190i)11-s + (0.534 − 0.217i)12-s + 0.831i·13-s − 0.633i·15-s + (−0.962 − 0.269i)16-s + (−0.385 + 0.222i)17-s + (−0.0641 − 0.327i)18-s + (0.477 − 0.826i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0275076 - 0.0519384i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0275076 - 0.0519384i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.06 - 0.929i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2.12 + 1.22i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.09 - 0.632i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.99iT - 13T^{2} \) |
| 17 | \( 1 + (1.58 - 0.916i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 3.60i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.83 + 3.36i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9.42T + 29T^{2} \) |
| 31 | \( 1 + (4.71 + 8.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.75 - 6.50i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.08iT - 41T^{2} \) |
| 43 | \( 1 + 6.27iT - 43T^{2} \) |
| 47 | \( 1 + (3.67 - 6.36i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0358 - 0.0620i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.68 + 2.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.61 - 5.55i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.43 - 1.40i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (7.01 - 4.05i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.54 + 0.891i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.33T + 83T^{2} \) |
| 89 | \( 1 + (7.42 + 4.28i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.10iT - 97T^{2} \) |
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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
−10.14482160872279896691105055633, −9.326204807034909347972532227565, −8.638376308910456386884539732433, −7.84679247829786069845289748529, −7.09832878592391958935277483771, −5.86826890588910058280386565580, −4.75366321504984658966019683079, −3.95229300356596669970264349009, −2.09301726527845169181374843591, −0.03889521895759926771524186364,
1.81898852998396217384675932756, 3.22075939131399483301263029616, 3.79745326330214806134246108818, 5.57495849270624049764103953721, 7.02099172777039829913294263493, 7.65456360782682392528754017088, 8.215866088521984845378099748975, 9.248423282544637689155375138537, 10.20577812776898975923435304369, 11.04069067382467664551082992652