L(s) = 1 | + 1.71·3-s + 1.91·5-s − 0.564·7-s − 0.0509·9-s − 3.11·11-s − 3.67·13-s + 3.28·15-s + 7.94·17-s − 19-s − 0.969·21-s − 2.64·23-s − 1.34·25-s − 5.23·27-s − 1.09·29-s − 7.05·31-s − 5.35·33-s − 1.07·35-s − 9.27·37-s − 6.30·39-s + 1.46·41-s − 4.03·43-s − 0.0974·45-s − 1.82·47-s − 6.68·49-s + 13.6·51-s − 13.3·53-s − 5.95·55-s + ⋯ |
L(s) = 1 | + 0.991·3-s + 0.855·5-s − 0.213·7-s − 0.0169·9-s − 0.939·11-s − 1.01·13-s + 0.847·15-s + 1.92·17-s − 0.229·19-s − 0.211·21-s − 0.552·23-s − 0.268·25-s − 1.00·27-s − 0.203·29-s − 1.26·31-s − 0.931·33-s − 0.182·35-s − 1.52·37-s − 1.01·39-s + 0.229·41-s − 0.614·43-s − 0.0145·45-s − 0.266·47-s − 0.954·49-s + 1.91·51-s − 1.83·53-s − 0.803·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 + 0.564T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 7.94T + 17T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 + 9.27T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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
−7.927935904068387039025570779619, −7.21364420237774931400700348437, −6.26900992806193274656546614887, −5.37675650243703533434819598712, −5.15489564519766455959867259683, −3.71047296361990014231617477247, −3.18012811002099138260128183804, −2.32675395686153488371842427926, −1.70789502515634530637895379824, 0,
1.70789502515634530637895379824, 2.32675395686153488371842427926, 3.18012811002099138260128183804, 3.71047296361990014231617477247, 5.15489564519766455959867259683, 5.37675650243703533434819598712, 6.26900992806193274656546614887, 7.21364420237774931400700348437, 7.927935904068387039025570779619