| L(s) = 1 | + 0.445·2-s − 0.0589·3-s − 1.80·4-s + 2.46·5-s − 0.0263·6-s − 2.81·7-s − 1.69·8-s − 2.99·9-s + 1.10·10-s + 4.28·11-s + 0.106·12-s + 4.41·13-s − 1.25·14-s − 0.145·15-s + 2.84·16-s + 6.60·17-s − 1.33·18-s + 4.20·19-s − 4.44·20-s + 0.166·21-s + 1.91·22-s + 23-s + 0.0999·24-s + 1.09·25-s + 1.97·26-s + 0.353·27-s + 5.07·28-s + ⋯ |
| L(s) = 1 | + 0.315·2-s − 0.0340·3-s − 0.900·4-s + 1.10·5-s − 0.0107·6-s − 1.06·7-s − 0.599·8-s − 0.998·9-s + 0.348·10-s + 1.29·11-s + 0.0306·12-s + 1.22·13-s − 0.335·14-s − 0.0376·15-s + 0.711·16-s + 1.60·17-s − 0.314·18-s + 0.965·19-s − 0.994·20-s + 0.0362·21-s + 0.407·22-s + 0.208·23-s + 0.0204·24-s + 0.219·25-s + 0.386·26-s + 0.0680·27-s + 0.958·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609282524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609282524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 3 | \( 1 + 0.0589T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 0.401T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 7.37T + 53T^{2} \) |
| 59 | \( 1 + 1.99T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 6.86T + 71T^{2} \) |
| 73 | \( 1 - 3.98T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 - 3.57T + 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 97 | \( 1 - 7.56T + 97T^{2} \) |
| 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
−10.19859602534041644654345232442, −9.558640395463344718312847223949, −9.055186206453195120451463563681, −8.131380828129404627525291344290, −6.55551254430624541934565236495, −5.91987420063444198029161517913, −5.31166300846453780180829796434, −3.72903076965046263038643049971, −3.12063516028778860917071593733, −1.13491634484006525420018702370,
1.13491634484006525420018702370, 3.12063516028778860917071593733, 3.72903076965046263038643049971, 5.31166300846453780180829796434, 5.91987420063444198029161517913, 6.55551254430624541934565236495, 8.131380828129404627525291344290, 9.055186206453195120451463563681, 9.558640395463344718312847223949, 10.19859602534041644654345232442
