L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2.56·7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 3.12·13-s + 2.56·14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s − 20-s + 2.56·21-s − 22-s + 6.56·23-s + 24-s + 25-s − 3.12·26-s − 27-s − 2.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.968·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.866·13-s + 0.684·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.558·21-s − 0.213·22-s + 1.36·23-s + 0.204·24-s + 0.200·25-s − 0.612·26-s − 0.192·27-s − 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6886048185\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6886048185\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2.56T + 7T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.68T + 97T^{2} \) |
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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
−8.357841920337290323968175477710, −7.26741365649908379382846829192, −6.74993415080249879938369858306, −6.24717543828721058977862492233, −5.43452172987535784232171198628, −4.41134255697790960017042244564, −3.61484810425167932274722406702, −2.82530654605725307722853002095, −1.58097132559433454281580946220, −0.51895233004753210864036658985,
0.51895233004753210864036658985, 1.58097132559433454281580946220, 2.82530654605725307722853002095, 3.61484810425167932274722406702, 4.41134255697790960017042244564, 5.43452172987535784232171198628, 6.24717543828721058977862492233, 6.74993415080249879938369858306, 7.26741365649908379382846829192, 8.357841920337290323968175477710