| L(s) = 1 | − 4.24·5-s − 2.13·7-s − 3.35·11-s + 0.0609·17-s + 6.18·19-s + 5.09·23-s + 13.0·25-s + 2.13·29-s − 2.85·31-s + 9.07·35-s + 3.47·37-s − 5.39·41-s − 8.61·43-s + 6.96·47-s − 2.43·49-s − 0.0217·53-s + 14.2·55-s + 5.81·59-s + 10.6·61-s − 7.40·67-s − 0.0489·71-s + 4.51·73-s + 7.17·77-s + 12.8·79-s − 3.55·83-s − 0.259·85-s − 2.34·89-s + ⋯ |
| L(s) = 1 | − 1.89·5-s − 0.807·7-s − 1.01·11-s + 0.0147·17-s + 1.41·19-s + 1.06·23-s + 2.60·25-s + 0.396·29-s − 0.512·31-s + 1.53·35-s + 0.570·37-s − 0.842·41-s − 1.31·43-s + 1.01·47-s − 0.347·49-s − 0.00299·53-s + 1.92·55-s + 0.756·59-s + 1.36·61-s − 0.904·67-s − 0.00580·71-s + 0.528·73-s + 0.817·77-s + 1.44·79-s − 0.390·83-s − 0.0280·85-s − 0.248·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 17 | \( 1 - 0.0609T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 2.13T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 + 5.39T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 + 0.0217T + 53T^{2} \) |
| 59 | \( 1 - 5.81T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 0.0489T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
| 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
−7.66794622019928099885085636847, −7.16359457025894420414859216295, −6.54906444865480830849000796656, −5.32395482375635407678684851153, −4.88292760695314809205260694438, −3.84685643462320700303897576237, −3.30525005687846425760570578564, −2.68795206829760224470768404756, −0.960078434252476594797161688188, 0,
0.960078434252476594797161688188, 2.68795206829760224470768404756, 3.30525005687846425760570578564, 3.84685643462320700303897576237, 4.88292760695314809205260694438, 5.32395482375635407678684851153, 6.54906444865480830849000796656, 7.16359457025894420414859216295, 7.66794622019928099885085636847
