L(s) = 1 | + 2.23i·5-s + 2.71i·7-s + 15.0i·11-s + 25.7·13-s + 20.2i·17-s + 26.1i·19-s + (22.4 − 5.18i)23-s − 5.00·25-s + 14.1·29-s + 28.7·31-s − 6.07·35-s − 1.93i·37-s + 18.2·41-s + 39.5i·43-s − 14.8·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.388i·7-s + 1.36i·11-s + 1.98·13-s + 1.18i·17-s + 1.37i·19-s + (0.974 − 0.225i)23-s − 0.200·25-s + 0.486·29-s + 0.926·31-s − 0.173·35-s − 0.0522i·37-s + 0.445·41-s + 0.919i·43-s − 0.315·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.682513754\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.682513754\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-22.4 + 5.18i)T \) |
good | 7 | \( 1 - 2.71iT - 49T^{2} \) |
| 11 | \( 1 - 15.0iT - 121T^{2} \) |
| 13 | \( 1 - 25.7T + 169T^{2} \) |
| 17 | \( 1 - 20.2iT - 289T^{2} \) |
| 19 | \( 1 - 26.1iT - 361T^{2} \) |
| 29 | \( 1 - 14.1T + 841T^{2} \) |
| 31 | \( 1 - 28.7T + 961T^{2} \) |
| 37 | \( 1 + 1.93iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 39.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 14.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 24.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 48.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 76.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 107.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 54.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 65.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 61.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 146. iT - 9.40e3T^{2} \) |
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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
−8.354110434174709767199816869602, −7.88366815433552674565944779865, −6.86169640981180216021318125283, −6.24815404176230389084840777381, −5.71153673175068392243859382676, −4.57436827672088860490254272306, −3.88194898889667035203057450731, −3.07966568660081917684145490111, −1.95096102267310078582167099246, −1.20766491750233404790813579569,
0.71353043758976719908739061572, 1.03856654347364932163549615796, 2.65255608828190520573988964429, 3.37999090375116784339711876520, 4.20948248799681995229547687257, 5.09930867717913993704495405366, 5.80870133503974988808657973014, 6.57436120228166085794344146943, 7.24670350362848448518266213602, 8.234641140763634781477704625376