| L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s − 3·11-s − 2i·13-s + 14-s + 16-s − 3i·17-s + 7·19-s + 3i·22-s − 2·26-s − i·28-s − 6·29-s − 4·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s − 0.904·11-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.727i·17-s + 1.60·19-s + 0.639i·22-s − 0.392·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9810938421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9810938421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 5iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| 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
−8.422744676750707553716049824440, −7.68023298155020637923512157752, −7.07982932957253178256932905795, −5.67219923136630603838785016438, −5.39430893478400508669661365282, −4.44876803856771021257999655318, −3.28397378274944978768731934921, −2.78582662491447571525563785209, −1.65270176310480196297829155039, −0.32523492560136225421712528518,
1.21157222186398183894077341966, 2.55444404414442935285995910406, 3.68302071037248763598897225823, 4.37657507233285635145635599896, 5.46866095553103841793803729044, 5.78708378126927330382522670052, 6.94419783847639042748313663024, 7.50317155980427810880870642999, 8.001046879665573109320359018896, 9.036394571262452559748210518960
