L(s) = 1 | − 2-s + (0.707 − 0.707i)3-s + 4-s + (−0.891 − 2.05i)5-s + (−0.707 + 0.707i)6-s + (−1.56 + 1.56i)7-s − 8-s − 1.00i·9-s + (0.891 + 2.05i)10-s − 3.92i·11-s + (0.707 − 0.707i)12-s − 0.319·13-s + (1.56 − 1.56i)14-s + (−2.08 − 0.819i)15-s + 16-s − 1.25i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.398 − 0.917i)5-s + (−0.288 + 0.288i)6-s + (−0.591 + 0.591i)7-s − 0.353·8-s − 0.333i·9-s + (0.282 + 0.648i)10-s − 1.18i·11-s + (0.204 − 0.204i)12-s − 0.0885·13-s + (0.418 − 0.418i)14-s + (−0.537 − 0.211i)15-s + 0.250·16-s − 0.304i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4477014545\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4477014545\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.891 + 2.05i)T \) |
| 37 | \( 1 + (6.08 + 0.0100i)T \) |
good | 7 | \( 1 + (1.56 - 1.56i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.92iT - 11T^{2} \) |
| 13 | \( 1 + 0.319T + 13T^{2} \) |
| 17 | \( 1 + 1.25iT - 17T^{2} \) |
| 19 | \( 1 + (-2.17 - 2.17i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.15T + 23T^{2} \) |
| 29 | \( 1 + (1.22 - 1.22i)T - 29iT^{2} \) |
| 31 | \( 1 + (6.78 + 6.78i)T + 31iT^{2} \) |
| 41 | \( 1 + 3.92iT - 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 + (4.88 - 4.88i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.03 + 3.03i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.70 - 4.70i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.56 + 5.56i)T + 61iT^{2} \) |
| 67 | \( 1 + (-6.88 - 6.88i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + (8.53 - 8.53i)T - 73iT^{2} \) |
| 79 | \( 1 + (3.59 + 3.59i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.13 + 1.13i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.33 + 1.33i)T - 89iT^{2} \) |
| 97 | \( 1 + 4.01iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.209839182982906263644242451354, −8.668255787349911739888263585672, −7.966458219159052913579432746777, −7.18886682128206650390160300069, −6.03869547651018991589989797079, −5.39901105845528228085057612196, −3.85354718708823515619239049716, −2.95035793655674615849535379096, −1.60474971943282737160485155821, −0.22815743327069628868051819372,
1.89934969479038953709704427807, 3.10514569833017167960609337408, 3.84889240197311568960808528160, 5.04247385244187540299714688176, 6.46359096114049168513422417858, 7.13868416893669774493831382164, 7.66375649515146420620853276957, 8.711852537418441285578396962314, 9.610853848805360717332550911901, 10.17010483027956954713021140045