| L(s) = 1 | + 2.00·2-s + 0.480·3-s + 2.00·4-s − 3.16·5-s + 0.961·6-s − 1.56·7-s + 0.00827·8-s − 2.76·9-s − 6.32·10-s − 0.984·11-s + 0.963·12-s + 0.535·13-s − 3.13·14-s − 1.51·15-s − 3.99·16-s − 4.27·17-s − 5.54·18-s + 4.66·19-s − 6.33·20-s − 0.751·21-s − 1.96·22-s + 1.91·23-s + 0.00397·24-s + 4.99·25-s + 1.07·26-s − 2.77·27-s − 3.13·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.277·3-s + 1.00·4-s − 1.41·5-s + 0.392·6-s − 0.591·7-s + 0.00292·8-s − 0.923·9-s − 2.00·10-s − 0.296·11-s + 0.277·12-s + 0.148·13-s − 0.836·14-s − 0.392·15-s − 0.997·16-s − 1.03·17-s − 1.30·18-s + 1.06·19-s − 1.41·20-s − 0.164·21-s − 0.419·22-s + 0.400·23-s + 0.000811·24-s + 0.999·25-s + 0.210·26-s − 0.533·27-s − 0.592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 3 | \( 1 - 0.480T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 0.984T + 11T^{2} \) |
| 13 | \( 1 - 0.535T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 23 | \( 1 - 1.91T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 0.998T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 1.90T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 1.02T + 97T^{2} \) |
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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
−9.406380826386049657927766179348, −8.714168975020169531037996156716, −7.71596302677961137614423148403, −6.94216479062415933818848485172, −5.91730251248923493997463904752, −5.07918264098068352208469524521, −4.04843972754989690258830031851, −3.41726454167295623796291998178, −2.61184006874969049991856194254, 0,
2.61184006874969049991856194254, 3.41726454167295623796291998178, 4.04843972754989690258830031851, 5.07918264098068352208469524521, 5.91730251248923493997463904752, 6.94216479062415933818848485172, 7.71596302677961137614423148403, 8.714168975020169531037996156716, 9.406380826386049657927766179348
