| L(s) = 1 | + 0.618·3-s + 5-s − 1.61·7-s − 2.61·9-s − 3.85·11-s + 4.09·13-s + 0.618·15-s − 5.09·17-s + 4.85·19-s − 1.00·21-s − 23-s + 25-s − 3.47·27-s − 4.76·29-s + 2.09·31-s − 2.38·33-s − 1.61·35-s − 2.47·37-s + 2.52·39-s − 12.3·41-s − 2.61·45-s − 9.70·47-s − 4.38·49-s − 3.14·51-s − 8.47·53-s − 3.85·55-s + 3.00·57-s + ⋯ |
| L(s) = 1 | + 0.356·3-s + 0.447·5-s − 0.611·7-s − 0.872·9-s − 1.16·11-s + 1.13·13-s + 0.159·15-s − 1.23·17-s + 1.11·19-s − 0.218·21-s − 0.208·23-s + 0.200·25-s − 0.668·27-s − 0.884·29-s + 0.375·31-s − 0.414·33-s − 0.273·35-s − 0.406·37-s + 0.404·39-s − 1.92·41-s − 0.390·45-s − 1.41·47-s − 0.625·49-s − 0.440·51-s − 1.16·53-s − 0.519·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.699434749526360806222975886583, −8.337931721751252323402596797343, −7.27970698667016167621697858886, −6.38680272369479985109405182628, −5.67981252570599653848758504594, −4.89146941254222277346751527699, −3.52668189144133905567600257059, −2.91493048445896874514802456851, −1.81529603433925383725961295886, 0,
1.81529603433925383725961295886, 2.91493048445896874514802456851, 3.52668189144133905567600257059, 4.89146941254222277346751527699, 5.67981252570599653848758504594, 6.38680272369479985109405182628, 7.27970698667016167621697858886, 8.337931721751252323402596797343, 8.699434749526360806222975886583
