| L(s) = 1 | + 2-s + 0.647·3-s + 4-s + 0.827·5-s + 0.647·6-s − 3.18·7-s + 8-s − 2.58·9-s + 0.827·10-s + 1.45·11-s + 0.647·12-s − 1.86·13-s − 3.18·14-s + 0.535·15-s + 16-s + 1.57·17-s − 2.58·18-s + 4.68·19-s + 0.827·20-s − 2.05·21-s + 1.45·22-s − 5.59·23-s + 0.647·24-s − 4.31·25-s − 1.86·26-s − 3.61·27-s − 3.18·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.373·3-s + 0.5·4-s + 0.369·5-s + 0.264·6-s − 1.20·7-s + 0.353·8-s − 0.860·9-s + 0.261·10-s + 0.437·11-s + 0.186·12-s − 0.516·13-s − 0.850·14-s + 0.138·15-s + 0.250·16-s + 0.381·17-s − 0.608·18-s + 1.07·19-s + 0.184·20-s − 0.449·21-s + 0.309·22-s − 1.16·23-s + 0.132·24-s − 0.863·25-s − 0.365·26-s − 0.695·27-s − 0.601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 0.647T + 3T^{2} \) |
| 5 | \( 1 - 0.827T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 5.59T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 - 0.922T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 9.71T + 59T^{2} \) |
| 61 | \( 1 - 0.979T + 61T^{2} \) |
| 67 | \( 1 - 5.64T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 0.218T + 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.88097061379671241380492725414, −7.31934724855720012034393548312, −6.42738629248458655450305237796, −5.76225544962635882340033292155, −5.32414997711798807035098459710, −3.98984345850215123141205486256, −3.44436249487688712694409833333, −2.69337664066314507678644999923, −1.75822736523109834469903874069, 0,
1.75822736523109834469903874069, 2.69337664066314507678644999923, 3.44436249487688712694409833333, 3.98984345850215123141205486256, 5.32414997711798807035098459710, 5.76225544962635882340033292155, 6.42738629248458655450305237796, 7.31934724855720012034393548312, 7.88097061379671241380492725414
