L(s) = 1 | + 1.22·2-s + 2.28·3-s − 0.507·4-s + 2.79·6-s + 1.28·7-s − 3.06·8-s + 2.22·9-s + 0.285·11-s − 1.15·12-s − 5·13-s + 1.57·14-s − 2.72·16-s + 6.23·17-s + 2.71·18-s + 2.93·21-s + 0.348·22-s + 5.23·23-s − 7.00·24-s − 6.10·26-s − 1.77·27-s − 0.651·28-s − 1.28·29-s + 1.22·31-s + 2.79·32-s + 0.651·33-s + 7.61·34-s − 1.12·36-s + ⋯ |
L(s) = 1 | + 0.863·2-s + 1.31·3-s − 0.253·4-s + 1.13·6-s + 0.485·7-s − 1.08·8-s + 0.740·9-s + 0.0859·11-s − 0.334·12-s − 1.38·13-s + 0.419·14-s − 0.682·16-s + 1.51·17-s + 0.639·18-s + 0.640·21-s + 0.0742·22-s + 1.09·23-s − 1.42·24-s − 1.19·26-s − 0.342·27-s − 0.123·28-s − 0.238·29-s + 0.219·31-s + 0.493·32-s + 0.113·33-s + 1.30·34-s − 0.187·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.651008126\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.651008126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 - 0.285T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 0.841T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 5.72T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 + 0.985T + 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 - 0.764T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 1.66T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.79058049445732013317371420272, −7.25925557931517146436889409193, −6.24930388399633763035815927256, −5.40338298601629609071790711714, −4.89376164278347901404634319820, −4.17304330550342576888938789203, −3.40198203981023653644326089741, −2.84330891131640788476667724823, −2.17087887967364315871950223725, −0.859670514604752430783571246779,
0.859670514604752430783571246779, 2.17087887967364315871950223725, 2.84330891131640788476667724823, 3.40198203981023653644326089741, 4.17304330550342576888938789203, 4.89376164278347901404634319820, 5.40338298601629609071790711714, 6.24930388399633763035815927256, 7.25925557931517146436889409193, 7.79058049445732013317371420272