| L(s) = 1 | + 1.83·2-s − 3-s + 1.36·4-s − 0.0163·5-s − 1.83·6-s − 2.68·7-s − 1.16·8-s + 9-s − 0.0300·10-s − 3.59·11-s − 1.36·12-s + 6.77·13-s − 4.92·14-s + 0.0163·15-s − 4.86·16-s + 17-s + 1.83·18-s − 1.16·19-s − 0.0223·20-s + 2.68·21-s − 6.59·22-s + 6.89·23-s + 1.16·24-s − 4.99·25-s + 12.4·26-s − 27-s − 3.66·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.682·4-s − 0.00731·5-s − 0.748·6-s − 1.01·7-s − 0.411·8-s + 0.333·9-s − 0.00949·10-s − 1.08·11-s − 0.394·12-s + 1.87·13-s − 1.31·14-s + 0.00422·15-s − 1.21·16-s + 0.242·17-s + 0.432·18-s − 0.266·19-s − 0.00499·20-s + 0.585·21-s − 1.40·22-s + 1.43·23-s + 0.237·24-s − 0.999·25-s + 2.43·26-s − 0.192·27-s − 0.692·28-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 | \( 1 - 1.83T + 2T^{2} \) |
| 5 | \( 1 + 0.0163T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 6.77T + 13T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 0.285T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 - 3.09T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 1.46T + 71T^{2} \) |
| 73 | \( 1 + 8.03T + 73T^{2} \) |
| 79 | \( 1 + 0.273T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 4.01T + 97T^{2} \) |
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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
−7.13868459967106938701968937423, −6.40630074346326207691664396544, −6.02641072943484614098907010784, −5.46999280961097208270629826718, −4.63801205265781447671359330135, −4.03047857759561635298137527159, −3.15387730081411375128872642731, −2.76606303161764080282159443047, −1.26331447072244048479518474510, 0,
1.26331447072244048479518474510, 2.76606303161764080282159443047, 3.15387730081411375128872642731, 4.03047857759561635298137527159, 4.63801205265781447671359330135, 5.46999280961097208270629826718, 6.02641072943484614098907010784, 6.40630074346326207691664396544, 7.13868459967106938701968937423
