L(s) = 1 | − 2-s − 3.34·3-s + 4-s − 3.64·5-s + 3.34·6-s + 5.09·7-s − 8-s + 8.19·9-s + 3.64·10-s − 0.225·11-s − 3.34·12-s − 4.05·13-s − 5.09·14-s + 12.1·15-s + 16-s + 0.866·17-s − 8.19·18-s − 0.400·19-s − 3.64·20-s − 17.0·21-s + 0.225·22-s + 2.97·23-s + 3.34·24-s + 8.28·25-s + 4.05·26-s − 17.3·27-s + 5.09·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.93·3-s + 0.5·4-s − 1.62·5-s + 1.36·6-s + 1.92·7-s − 0.353·8-s + 2.73·9-s + 1.15·10-s − 0.0680·11-s − 0.965·12-s − 1.12·13-s − 1.36·14-s + 3.14·15-s + 0.250·16-s + 0.210·17-s − 1.93·18-s − 0.0918·19-s − 0.814·20-s − 3.71·21-s + 0.0481·22-s + 0.620·23-s + 0.682·24-s + 1.65·25-s + 0.795·26-s − 3.34·27-s + 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 + 0.225T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 - 0.866T + 17T^{2} \) |
| 19 | \( 1 + 0.400T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 - 5.14T + 37T^{2} \) |
| 41 | \( 1 + 4.50T + 41T^{2} \) |
| 43 | \( 1 - 5.90T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.83T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 + 0.790T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 + 9.99T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 - 8.02T + 79T^{2} \) |
| 83 | \( 1 - 3.91T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.73723281083021212765390842041, −7.48341964553768715636909598408, −6.91293641841255288215265382032, −5.75570093720847235276603070886, −4.93891581650094005540781874936, −4.66806375643970093390411464868, −3.71616727259450359699139455813, −1.96445047750717731741727601277, −0.968193311901888109980787467137, 0,
0.968193311901888109980787467137, 1.96445047750717731741727601277, 3.71616727259450359699139455813, 4.66806375643970093390411464868, 4.93891581650094005540781874936, 5.75570093720847235276603070886, 6.91293641841255288215265382032, 7.48341964553768715636909598408, 7.73723281083021212765390842041