| L(s) = 1 | − 2-s − 1.75·3-s + 4-s − 0.498·5-s + 1.75·6-s − 5.22·7-s − 8-s + 0.0865·9-s + 0.498·10-s + 11-s − 1.75·12-s − 3.41·13-s + 5.22·14-s + 0.875·15-s + 16-s + 3.49·17-s − 0.0865·18-s − 0.498·20-s + 9.18·21-s − 22-s − 1.11·23-s + 1.75·24-s − 4.75·25-s + 3.41·26-s + 5.11·27-s − 5.22·28-s + 4.95·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.01·3-s + 0.5·4-s − 0.222·5-s + 0.717·6-s − 1.97·7-s − 0.353·8-s + 0.0288·9-s + 0.157·10-s + 0.301·11-s − 0.507·12-s − 0.947·13-s + 1.39·14-s + 0.226·15-s + 0.250·16-s + 0.847·17-s − 0.0203·18-s − 0.111·20-s + 2.00·21-s − 0.213·22-s − 0.232·23-s + 0.358·24-s − 0.950·25-s + 0.670·26-s + 0.985·27-s − 0.988·28-s + 0.920·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 0.498T + 5T^{2} \) |
| 7 | \( 1 + 5.22T + 7T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 - 4.95T + 29T^{2} \) |
| 31 | \( 1 - 0.897T + 31T^{2} \) |
| 37 | \( 1 + 6.56T + 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 3.31T + 61T^{2} \) |
| 67 | \( 1 - 9.03T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 7.09T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−7.26920851390091985946397424912, −6.76422167040768293328224595241, −6.28548105783320383492753477580, −5.60186144314548876208934683071, −4.90157012114189553832280333496, −3.64692980762906127674450663928, −3.17763308369822639514140423802, −2.16465442498387007937703134292, −0.73315323919179652700640631540, 0,
0.73315323919179652700640631540, 2.16465442498387007937703134292, 3.17763308369822639514140423802, 3.64692980762906127674450663928, 4.90157012114189553832280333496, 5.60186144314548876208934683071, 6.28548105783320383492753477580, 6.76422167040768293328224595241, 7.26920851390091985946397424912
