| L(s) = 1 | − 2.40·2-s + 2.42·3-s + 3.77·4-s + 1.57·5-s − 5.82·6-s − 3.75·7-s − 4.27·8-s + 2.86·9-s − 3.79·10-s + 2.83·11-s + 9.15·12-s + 2.38·13-s + 9.01·14-s + 3.82·15-s + 2.71·16-s − 1.20·17-s − 6.89·18-s − 2.50·19-s + 5.96·20-s − 9.08·21-s − 6.82·22-s − 6.40·23-s − 10.3·24-s − 2.50·25-s − 5.74·26-s − 0.322·27-s − 14.1·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.39·3-s + 1.88·4-s + 0.705·5-s − 2.37·6-s − 1.41·7-s − 1.51·8-s + 0.955·9-s − 1.19·10-s + 0.856·11-s + 2.64·12-s + 0.662·13-s + 2.40·14-s + 0.987·15-s + 0.678·16-s − 0.292·17-s − 1.62·18-s − 0.574·19-s + 1.33·20-s − 1.98·21-s − 1.45·22-s − 1.33·23-s − 2.11·24-s − 0.501·25-s − 1.12·26-s − 0.0621·27-s − 2.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 - 2.42T + 3T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 - 2.83T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 6.40T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 - 2.39T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 8.99T + 97T^{2} \) |
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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
−7.84903677856839200049355631429, −6.89339666085820440958663757674, −6.42018701267152378672105440847, −5.98936098258996787945422400500, −4.34281319912290541500896835949, −3.51248519358082656901147243467, −2.80157629881630287636951250161, −2.07445967560423732265752026654, −1.35512972405529072124820713518, 0,
1.35512972405529072124820713518, 2.07445967560423732265752026654, 2.80157629881630287636951250161, 3.51248519358082656901147243467, 4.34281319912290541500896835949, 5.98936098258996787945422400500, 6.42018701267152378672105440847, 6.89339666085820440958663757674, 7.84903677856839200049355631429
