| L(s) = 1 | + 0.330·2-s + 2.43·3-s − 1.89·4-s + 5-s + 0.805·6-s − 1.28·8-s + 2.92·9-s + 0.330·10-s − 11-s − 4.60·12-s − 0.0114·13-s + 2.43·15-s + 3.35·16-s + 3.37·17-s + 0.968·18-s − 0.436·19-s − 1.89·20-s − 0.330·22-s + 4.40·23-s − 3.13·24-s + 25-s − 0.00377·26-s − 0.173·27-s + 4.69·29-s + 0.805·30-s + 2.78·31-s + 3.68·32-s + ⋯ |
| L(s) = 1 | + 0.233·2-s + 1.40·3-s − 0.945·4-s + 0.447·5-s + 0.328·6-s − 0.454·8-s + 0.976·9-s + 0.104·10-s − 0.301·11-s − 1.32·12-s − 0.00316·13-s + 0.628·15-s + 0.838·16-s + 0.819·17-s + 0.228·18-s − 0.100·19-s − 0.422·20-s − 0.0705·22-s + 0.917·23-s − 0.639·24-s + 0.200·25-s − 0.000740·26-s − 0.0333·27-s + 0.871·29-s + 0.147·30-s + 0.501·31-s + 0.651·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.999717885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.999717885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.330T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 13 | \( 1 + 0.0114T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 + 0.436T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 0.00191T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 - 8.86T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 7.75T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.732965826716934111239671953434, −8.291871121082361655932250193301, −7.61023151538428900727122729534, −6.59208133362487539713202521135, −5.56037795509321122470168906952, −4.84795457039562179609531427854, −3.90645005097046885707285737365, −3.14740095094140419364334798554, −2.41033743434936246146341503668, −1.03472380820354845652203554438,
1.03472380820354845652203554438, 2.41033743434936246146341503668, 3.14740095094140419364334798554, 3.90645005097046885707285737365, 4.84795457039562179609531427854, 5.56037795509321122470168906952, 6.59208133362487539713202521135, 7.61023151538428900727122729534, 8.291871121082361655932250193301, 8.732965826716934111239671953434
