| L(s) = 1 | + 2-s + 4-s − 2.38·5-s + 1.61·7-s + 8-s − 2.38·10-s − 1.29·11-s − 6.29·13-s + 1.61·14-s + 16-s + 6.73·17-s − 4.10·19-s − 2.38·20-s − 1.29·22-s + 8.82·23-s + 0.677·25-s − 6.29·26-s + 1.61·28-s − 3.74·29-s + 2.10·31-s + 32-s + 6.73·34-s − 3.84·35-s − 9.37·37-s − 4.10·38-s − 2.38·40-s − 8.42·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.06·5-s + 0.609·7-s + 0.353·8-s − 0.753·10-s − 0.390·11-s − 1.74·13-s + 0.431·14-s + 0.250·16-s + 1.63·17-s − 0.941·19-s − 0.532·20-s − 0.275·22-s + 1.84·23-s + 0.135·25-s − 1.23·26-s + 0.304·28-s − 0.694·29-s + 0.377·31-s + 0.176·32-s + 1.15·34-s − 0.649·35-s − 1.54·37-s − 0.665·38-s − 0.376·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 2.10T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 3.32T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 + 7.88T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 4.37T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 - 7.03T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.008963575761621289097373140156, −7.20035685190131497420655058385, −6.90599939516568157039641722759, −5.45004453513696317635018713083, −5.09953041377310457369382716618, −4.35672888197568198243809482881, −3.43642762080122250982860197875, −2.74070500089549069615216174380, −1.56051860456743620234937186886, 0,
1.56051860456743620234937186886, 2.74070500089549069615216174380, 3.43642762080122250982860197875, 4.35672888197568198243809482881, 5.09953041377310457369382716618, 5.45004453513696317635018713083, 6.90599939516568157039641722759, 7.20035685190131497420655058385, 8.008963575761621289097373140156
