L(s) = 1 | + 1.19·2-s − 3-s − 0.583·4-s − 1.54·5-s − 1.19·6-s − 7-s − 3.07·8-s + 9-s − 1.84·10-s − 0.938·11-s + 0.583·12-s + 1.40·13-s − 1.19·14-s + 1.54·15-s − 2.49·16-s + 2.57·17-s + 1.19·18-s + 4.76·19-s + 0.901·20-s + 21-s − 1.11·22-s − 3.41·23-s + 3.07·24-s − 2.61·25-s + 1.66·26-s − 27-s + 0.583·28-s + ⋯ |
L(s) = 1 | + 0.841·2-s − 0.577·3-s − 0.291·4-s − 0.691·5-s − 0.485·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s − 0.581·10-s − 0.282·11-s + 0.168·12-s + 0.388·13-s − 0.318·14-s + 0.399·15-s − 0.623·16-s + 0.625·17-s + 0.280·18-s + 1.09·19-s + 0.201·20-s + 0.218·21-s − 0.238·22-s − 0.711·23-s + 0.627·24-s − 0.522·25-s + 0.327·26-s − 0.192·27-s + 0.110·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 11 | \( 1 + 0.938T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 0.725T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + 2.08T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 6.52T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 9.85T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 4.55T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 9.15T + 97T^{2} \) |
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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
−7.35624669812629875875104710068, −6.66476712496930683112991850018, −5.82561552989536479852895516429, −5.42057593786439714550858257587, −4.69423965184246546357317985756, −3.82849707366630977614440123214, −3.51290479670199020497190169495, −2.51035298985084518796464571705, −1.03986142580286039812289352963, 0,
1.03986142580286039812289352963, 2.51035298985084518796464571705, 3.51290479670199020497190169495, 3.82849707366630977614440123214, 4.69423965184246546357317985756, 5.42057593786439714550858257587, 5.82561552989536479852895516429, 6.66476712496930683112991850018, 7.35624669812629875875104710068