| L(s) = 1 | − 2.17·2-s − 3-s + 2.74·4-s − 2.11·5-s + 2.17·6-s + 2.69·7-s − 1.63·8-s + 9-s + 4.60·10-s + 3.86·11-s − 2.74·12-s − 2.95·13-s − 5.87·14-s + 2.11·15-s − 1.94·16-s − 17-s − 2.17·18-s + 2.38·19-s − 5.80·20-s − 2.69·21-s − 8.42·22-s + 6.54·23-s + 1.63·24-s − 0.539·25-s + 6.44·26-s − 27-s + 7.41·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.577·3-s + 1.37·4-s − 0.944·5-s + 0.889·6-s + 1.01·7-s − 0.576·8-s + 0.333·9-s + 1.45·10-s + 1.16·11-s − 0.793·12-s − 0.820·13-s − 1.57·14-s + 0.545·15-s − 0.485·16-s − 0.242·17-s − 0.513·18-s + 0.546·19-s − 1.29·20-s − 0.588·21-s − 1.79·22-s + 1.36·23-s + 0.332·24-s − 0.107·25-s + 1.26·26-s − 0.192·27-s + 1.40·28-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 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 + 0.373T + 29T^{2} \) |
| 31 | \( 1 - 0.0561T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 + 8.21T + 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 3.35T + 73T^{2} \) |
| 79 | \( 1 + 9.02T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.44335916453311745316726202381, −7.21063005497497898879488129951, −6.48289337539386498914952638899, −5.39618349317116170933913941120, −4.63232472234054479776499324932, −4.05439596223498545488973813110, −2.86812475331440571900174425096, −1.69259279996982695353933793532, −1.06078946134184179725800490167, 0,
1.06078946134184179725800490167, 1.69259279996982695353933793532, 2.86812475331440571900174425096, 4.05439596223498545488973813110, 4.63232472234054479776499324932, 5.39618349317116170933913941120, 6.48289337539386498914952638899, 7.21063005497497898879488129951, 7.44335916453311745316726202381
