| L(s) = 1 | + 1.99·2-s − 3-s + 1.97·4-s + 1.60·5-s − 1.99·6-s − 0.432·7-s − 0.0559·8-s + 9-s + 3.19·10-s + 1.01·11-s − 1.97·12-s − 6.62·13-s − 0.862·14-s − 1.60·15-s − 4.05·16-s − 17-s + 1.99·18-s + 1.52·19-s + 3.15·20-s + 0.432·21-s + 2.01·22-s + 4.61·23-s + 0.0559·24-s − 2.43·25-s − 13.2·26-s − 27-s − 0.853·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.985·4-s + 0.716·5-s − 0.813·6-s − 0.163·7-s − 0.0197·8-s + 0.333·9-s + 1.00·10-s + 0.305·11-s − 0.569·12-s − 1.83·13-s − 0.230·14-s − 0.413·15-s − 1.01·16-s − 0.242·17-s + 0.469·18-s + 0.349·19-s + 0.705·20-s + 0.0944·21-s + 0.429·22-s + 0.962·23-s + 0.0114·24-s − 0.487·25-s − 2.58·26-s − 0.192·27-s − 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 + 0.432T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 2.02T + 37T^{2} \) |
| 41 | \( 1 + 0.00303T + 41T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 4.49T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 1.11T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 - 0.643T + 97T^{2} \) |
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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
−7.09883843523671599059579701036, −6.51928664814853930370557964819, −6.04045230640758047527755094897, −5.13018699027931906255052274439, −4.87657443053649270788425826079, −4.20069555457518994389141988448, −3.04400190206853359435078625686, −2.60428881086815979415618771575, −1.50440359373712454239419171791, 0,
1.50440359373712454239419171791, 2.60428881086815979415618771575, 3.04400190206853359435078625686, 4.20069555457518994389141988448, 4.87657443053649270788425826079, 5.13018699027931906255052274439, 6.04045230640758047527755094897, 6.51928664814853930370557964819, 7.09883843523671599059579701036
