L(s) = 1 | − 2-s + 0.357·3-s − 4-s + 2.58·5-s − 0.357·6-s − 0.357·7-s + 3·8-s − 2.87·9-s − 2.58·10-s − 0.357·12-s − 4.94·13-s + 0.357·14-s + 0.926·15-s − 16-s − 17-s + 2.87·18-s − 2.35·19-s − 2.58·20-s − 0.128·21-s − 2.94·23-s + 1.07·24-s + 1.69·25-s + 4.94·26-s − 2.10·27-s + 0.357·28-s + 3.30·29-s − 0.926·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.206·3-s − 0.5·4-s + 1.15·5-s − 0.146·6-s − 0.135·7-s + 1.06·8-s − 0.957·9-s − 0.818·10-s − 0.103·12-s − 1.37·13-s + 0.0956·14-s + 0.239·15-s − 0.250·16-s − 0.242·17-s + 0.676·18-s − 0.540·19-s − 0.578·20-s − 0.0279·21-s − 0.614·23-s + 0.219·24-s + 0.339·25-s + 0.969·26-s − 0.404·27-s + 0.0676·28-s + 0.613·29-s − 0.169·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 - 0.357T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 + 0.357T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 9.15T + 41T^{2} \) |
| 43 | \( 1 + 6.22T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 5.41T + 53T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 + 2.56T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 + 0.715T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 8.45T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−9.490825593021134780005059297112, −8.910861493498650870279590960295, −8.112841247579290778046672616941, −7.20223309657210519655909044921, −6.07889668494302932202845313808, −5.28643071391978720838586596986, −4.32568189369840918380567417881, −2.80658392677841449392391161211, −1.82389888864526036834252196026, 0,
1.82389888864526036834252196026, 2.80658392677841449392391161211, 4.32568189369840918380567417881, 5.28643071391978720838586596986, 6.07889668494302932202845313808, 7.20223309657210519655909044921, 8.112841247579290778046672616941, 8.910861493498650870279590960295, 9.490825593021134780005059297112