| L(s) = 1 | + 1.61·5-s − 3.85·7-s + 2.38·13-s − 2.38·17-s + 3.85·19-s + 2.47·23-s − 2.38·25-s − 8.61·29-s + 0.854·31-s − 6.23·35-s − 1.85·37-s − 8.61·41-s − 1.38·47-s + 7.85·49-s + 4.09·53-s − 1.09·59-s + 2.38·61-s + 3.85·65-s + 12.9·67-s + 6.38·71-s − 0.909·73-s − 7.14·79-s − 13.0·83-s − 3.85·85-s − 0.472·89-s − 9.18·91-s + 6.23·95-s + ⋯ |
| L(s) = 1 | + 0.723·5-s − 1.45·7-s + 0.660·13-s − 0.577·17-s + 0.884·19-s + 0.515·23-s − 0.476·25-s − 1.60·29-s + 0.153·31-s − 1.05·35-s − 0.304·37-s − 1.34·41-s − 0.201·47-s + 1.12·49-s + 0.561·53-s − 0.141·59-s + 0.304·61-s + 0.478·65-s + 1.58·67-s + 0.757·71-s − 0.106·73-s − 0.803·79-s − 1.43·83-s − 0.418·85-s − 0.0500·89-s − 0.962·91-s + 0.639·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 - 0.854T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 + 8.61T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 + 1.09T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.909T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 0.472T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.061752574979440662322928413764, −6.97882899379567160572437508545, −6.66229631407951040189487139250, −5.74519379358932308061752559075, −5.32684827605482305094910056943, −4.00882133342716792405913109988, −3.37908160669909340816606638970, −2.50914578970101574220987541427, −1.43342325611455506712762512021, 0,
1.43342325611455506712762512021, 2.50914578970101574220987541427, 3.37908160669909340816606638970, 4.00882133342716792405913109988, 5.32684827605482305094910056943, 5.74519379358932308061752559075, 6.66229631407951040189487139250, 6.97882899379567160572437508545, 8.061752574979440662322928413764
