| L(s) = 1 | + 0.347·2-s − 1.87·3-s − 0.879·4-s − 0.652·6-s − 1.87·7-s − 0.652·8-s + 2.53·9-s + 1.65·12-s − 0.652·14-s + 0.652·16-s + 0.879·18-s + 0.347·19-s + 3.53·21-s + 1.53·23-s + 1.22·24-s + 25-s − 2.87·27-s + 1.65·28-s − 31-s + 0.879·32-s − 2.22·36-s − 37-s + 0.120·38-s − 41-s + 1.22·42-s + 0.347·43-s + 0.532·46-s + ⋯ |
| L(s) = 1 | + 0.347·2-s − 1.87·3-s − 0.879·4-s − 0.652·6-s − 1.87·7-s − 0.652·8-s + 2.53·9-s + 1.65·12-s − 0.652·14-s + 0.652·16-s + 0.879·18-s + 0.347·19-s + 3.53·21-s + 1.53·23-s + 1.22·24-s + 25-s − 2.87·27-s + 1.65·28-s − 31-s + 0.879·32-s − 2.22·36-s − 37-s + 0.120·38-s − 41-s + 1.22·42-s + 0.347·43-s + 0.532·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3644152765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3644152765\) |
| \(L(1)\) |
|
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 | \( 1 - 0.347T + T^{2} \) |
| 3 | \( 1 + 1.87T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.347T + T^{2} \) |
| 23 | \( 1 - 1.53T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - 0.347T + T^{2} \) |
| 47 | \( 1 - 1.53T + T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.87T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.29824685842766052219900405303, −9.545494624930765552546534024175, −8.877529265869158588887363360290, −7.09124974543818856966954821628, −6.71873057779584683151178845549, −5.66807011882561678570392744825, −5.25774182095775294148987189926, −4.15141427555663062534297735497, −3.21990718946503936114342347533, −0.72070521974543945454520510482,
0.72070521974543945454520510482, 3.21990718946503936114342347533, 4.15141427555663062534297735497, 5.25774182095775294148987189926, 5.66807011882561678570392744825, 6.71873057779584683151178845549, 7.09124974543818856966954821628, 8.877529265869158588887363360290, 9.545494624930765552546534024175, 10.29824685842766052219900405303
