| L(s) = 1 | − 2-s − 2.80·3-s + 4-s + 0.364·5-s + 2.80·6-s + 1.53·7-s − 8-s + 4.84·9-s − 0.364·10-s + 1.43·11-s − 2.80·12-s + 3.76·13-s − 1.53·14-s − 1.01·15-s + 16-s + 3.34·17-s − 4.84·18-s + 0.670·19-s + 0.364·20-s − 4.28·21-s − 1.43·22-s + 2.07·23-s + 2.80·24-s − 4.86·25-s − 3.76·26-s − 5.16·27-s + 1.53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.162·5-s + 1.14·6-s + 0.578·7-s − 0.353·8-s + 1.61·9-s − 0.115·10-s + 0.433·11-s − 0.808·12-s + 1.04·13-s − 0.409·14-s − 0.263·15-s + 0.250·16-s + 0.810·17-s − 1.14·18-s + 0.153·19-s + 0.0814·20-s − 0.935·21-s − 0.306·22-s + 0.432·23-s + 0.571·24-s − 0.973·25-s − 0.738·26-s − 0.993·27-s + 0.289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 0.364T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.670T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 1.76T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 + 4.25T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.74645209533389604193201455491, −6.82512446029740707131973770226, −6.39493166690532079908370887795, −5.57474899968481450969507824246, −5.19976726231098127646771730369, −4.15477718312146019080562453405, −3.28926641912631625917405920313, −1.72183175412472660134852714227, −1.21566045617250551215477725673, 0,
1.21566045617250551215477725673, 1.72183175412472660134852714227, 3.28926641912631625917405920313, 4.15477718312146019080562453405, 5.19976726231098127646771730369, 5.57474899968481450969507824246, 6.39493166690532079908370887795, 6.82512446029740707131973770226, 7.74645209533389604193201455491
