| L(s) = 1 | − 3.04·5-s − 5.04·7-s + 3.80·11-s + 0.356·17-s − 6.13·19-s + 7.93·23-s + 4.29·25-s − 1.41·29-s + 5.78·31-s + 15.3·35-s + 0.0271·37-s + 9.18·41-s − 1.21·43-s + 9.56·47-s + 18.4·49-s + 4.73·53-s − 11.5·55-s − 13.5·59-s + 0.576·61-s − 7.95·67-s + 3.07·71-s − 14.0·73-s − 19.1·77-s − 0.841·79-s + 0.192·83-s − 1.08·85-s − 9.21·89-s + ⋯ |
| L(s) = 1 | − 1.36·5-s − 1.90·7-s + 1.14·11-s + 0.0865·17-s − 1.40·19-s + 1.65·23-s + 0.859·25-s − 0.263·29-s + 1.03·31-s + 2.60·35-s + 0.00446·37-s + 1.43·41-s − 0.186·43-s + 1.39·47-s + 2.64·49-s + 0.650·53-s − 1.56·55-s − 1.76·59-s + 0.0738·61-s − 0.971·67-s + 0.365·71-s − 1.64·73-s − 2.18·77-s − 0.0946·79-s + 0.0211·83-s − 0.118·85-s − 0.976·89-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 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 17 | \( 1 - 0.356T + 17T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 0.0271T + 37T^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 0.576T + 61T^{2} \) |
| 67 | \( 1 + 7.95T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 0.841T + 79T^{2} \) |
| 83 | \( 1 - 0.192T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 + 0.682T + 97T^{2} \) |
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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
−7.53321824871438486671398148279, −6.99462821104612010901884328128, −6.42881477547968832529253141383, −5.81245891049363640066049011533, −4.43159022131840788035908704789, −4.06444510319973456994043443641, −3.27693864199093471942948575493, −2.65677668054898014742846885276, −0.997017769899252746805142115898, 0,
0.997017769899252746805142115898, 2.65677668054898014742846885276, 3.27693864199093471942948575493, 4.06444510319973456994043443641, 4.43159022131840788035908704789, 5.81245891049363640066049011533, 6.42881477547968832529253141383, 6.99462821104612010901884328128, 7.53321824871438486671398148279
