| L(s) = 1 | + 1.23·5-s − 7-s − 1.23·11-s + 4.47·13-s + 5.23·17-s − 6.47·19-s − 7.70·23-s − 3.47·25-s − 10.4·29-s − 6.47·31-s − 1.23·35-s − 4.47·37-s + 2.76·41-s + 4.94·43-s + 10.4·47-s + 49-s − 8·53-s − 1.52·55-s − 2.47·59-s − 4.47·61-s + 5.52·65-s − 8·67-s + 2.76·71-s + 2.94·73-s + 1.23·77-s + 4.94·79-s + 4.94·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s − 0.377·7-s − 0.372·11-s + 1.24·13-s + 1.26·17-s − 1.48·19-s − 1.60·23-s − 0.694·25-s − 1.94·29-s − 1.16·31-s − 0.208·35-s − 0.735·37-s + 0.431·41-s + 0.753·43-s + 1.52·47-s + 0.142·49-s − 1.09·53-s − 0.206·55-s − 0.321·59-s − 0.572·61-s + 0.685·65-s − 0.977·67-s + 0.328·71-s + 0.344·73-s + 0.140·77-s + 0.556·79-s + 0.542·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2.76T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.916162473679232942789243533910, −7.56370020460753611187900564391, −6.30578053484767896248865146068, −5.95911155641328511714779538168, −5.32781406935041049708948868184, −3.96654687167480479925891186254, −3.63803733212240302439789384969, −2.31980949802386837485865884137, −1.57469353209252980663596296546, 0,
1.57469353209252980663596296546, 2.31980949802386837485865884137, 3.63803733212240302439789384969, 3.96654687167480479925891186254, 5.32781406935041049708948868184, 5.95911155641328511714779538168, 6.30578053484767896248865146068, 7.56370020460753611187900564391, 7.916162473679232942789243533910
