| L(s) = 1 | + 5-s + 3.56·7-s − 4.12·11-s − 0.438·13-s − 5.56·17-s − 2·19-s + 3.68·23-s + 25-s + 29-s − 6.24·31-s + 3.56·35-s − 2.56·37-s − 9.68·41-s − 1.43·43-s − 8.43·47-s + 5.68·49-s + 0.561·53-s − 4.12·55-s − 7.12·59-s − 0.438·65-s + 10.6·67-s − 9.12·71-s − 1.43·73-s − 14.6·77-s − 7.12·79-s − 2.31·83-s − 5.56·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.34·7-s − 1.24·11-s − 0.121·13-s − 1.34·17-s − 0.458·19-s + 0.768·23-s + 0.200·25-s + 0.185·29-s − 1.12·31-s + 0.602·35-s − 0.421·37-s − 1.51·41-s − 0.219·43-s − 1.23·47-s + 0.812·49-s + 0.0771·53-s − 0.555·55-s − 0.927·59-s − 0.0543·65-s + 1.30·67-s − 1.08·71-s − 0.168·73-s − 1.67·77-s − 0.801·79-s − 0.254·83-s − 0.603·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 + 2.31T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 - 2.31T + 97T^{2} \) |
| 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
−7.935906779082870074216165508975, −7.15708625530888481602332554621, −6.48316586932269897294595578719, −5.42649586793818752817099810243, −4.99927098344645251721897683074, −4.36669112208241663861371499849, −3.15805223871441721361209348289, −2.21031554846628694386124218605, −1.59607646953145367989932721272, 0,
1.59607646953145367989932721272, 2.21031554846628694386124218605, 3.15805223871441721361209348289, 4.36669112208241663861371499849, 4.99927098344645251721897683074, 5.42649586793818752817099810243, 6.48316586932269897294595578719, 7.15708625530888481602332554621, 7.935906779082870074216165508975
