L(s) = 1 | − 0.951·5-s − 1.28·7-s + 0.951·11-s − 2.82·13-s − 1.31·17-s + 5.94·19-s + 2.56·23-s − 4.09·25-s + 2.48·29-s + 0.984·31-s + 1.21·35-s − 7.81·37-s + 7.41·41-s + 7.65·43-s − 10.1·47-s − 5.35·49-s + 3.23·53-s − 0.904·55-s + 10.3·59-s − 11.8·61-s + 2.68·65-s + 8.14·67-s − 1.71·71-s − 0.502·73-s − 1.21·77-s − 14.6·79-s + 11.3·83-s + ⋯ |
L(s) = 1 | − 0.425·5-s − 0.484·7-s + 0.286·11-s − 0.784·13-s − 0.318·17-s + 1.36·19-s + 0.535·23-s − 0.819·25-s + 0.462·29-s + 0.176·31-s + 0.206·35-s − 1.28·37-s + 1.15·41-s + 1.16·43-s − 1.47·47-s − 0.765·49-s + 0.444·53-s − 0.122·55-s + 1.34·59-s − 1.52·61-s + 0.333·65-s + 0.995·67-s − 0.203·71-s − 0.0587·73-s − 0.138·77-s − 1.65·79-s + 1.24·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 0.951T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 0.951T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 0.984T + 31T^{2} \) |
| 37 | \( 1 + 7.81T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 + 1.71T + 71T^{2} \) |
| 73 | \( 1 + 0.502T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 6.15T + 89T^{2} \) |
| 97 | \( 1 - 4.75T + 97T^{2} \) |
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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
−7.63355339256429353160511116505, −7.09781088912242508288642262145, −6.40031287491029302506406830312, −5.52674968432294625108468945511, −4.84867719900937316276912571721, −3.99389401966633799739567081359, −3.23885387181180400624779964981, −2.45739273861161578006893434248, −1.23292041884592317070890126957, 0,
1.23292041884592317070890126957, 2.45739273861161578006893434248, 3.23885387181180400624779964981, 3.99389401966633799739567081359, 4.84867719900937316276912571721, 5.52674968432294625108468945511, 6.40031287491029302506406830312, 7.09781088912242508288642262145, 7.63355339256429353160511116505