| L(s) = 1 | − 2-s − 3-s + 4-s − 2.93·5-s + 6-s − 7-s − 8-s + 9-s + 2.93·10-s − 2.81·11-s − 12-s + 14-s + 2.93·15-s + 16-s + 4.00·17-s − 18-s − 6.01·19-s − 2.93·20-s + 21-s + 2.81·22-s − 2.16·23-s + 24-s + 3.60·25-s − 27-s − 28-s + 7.58·29-s − 2.93·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.31·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.927·10-s − 0.847·11-s − 0.288·12-s + 0.267·14-s + 0.757·15-s + 0.250·16-s + 0.972·17-s − 0.235·18-s − 1.38·19-s − 0.656·20-s + 0.218·21-s + 0.599·22-s − 0.451·23-s + 0.204·24-s + 0.721·25-s − 0.192·27-s − 0.188·28-s + 1.40·29-s − 0.535·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 - 3.99T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - 1.79T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 2.80T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 - 8.84T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.64567506159910944482865629494, −7.07330343992495198869978035040, −6.29413275036415900756976481543, −5.61669131386123140323393500781, −4.67235162023819929210353892949, −3.96599193816035784291236901432, −3.14435507761957628733508401326, −2.20527047672282686035984061226, −0.844979153868344675175342802477, 0,
0.844979153868344675175342802477, 2.20527047672282686035984061226, 3.14435507761957628733508401326, 3.96599193816035784291236901432, 4.67235162023819929210353892949, 5.61669131386123140323393500781, 6.29413275036415900756976481543, 7.07330343992495198869978035040, 7.64567506159910944482865629494
