| L(s) = 1 | − 3·7-s − 3·11-s + 13-s − 7·17-s + 8·19-s + 4·23-s + 3·29-s + 11·31-s + 2·41-s + 8·43-s − 9·47-s + 2·49-s − 9·53-s − 9·59-s + 61-s − 5·67-s − 12·73-s + 9·77-s + 8·79-s − 9·83-s − 12·89-s − 3·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.904·11-s + 0.277·13-s − 1.69·17-s + 1.83·19-s + 0.834·23-s + 0.557·29-s + 1.97·31-s + 0.312·41-s + 1.21·43-s − 1.31·47-s + 2/7·49-s − 1.23·53-s − 1.17·59-s + 0.128·61-s − 0.610·67-s − 1.40·73-s + 1.02·77-s + 0.900·79-s − 0.987·83-s − 1.27·89-s − 0.314·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.97038575903942, −13.47370638772228, −13.12792140803411, −12.75267439282122, −12.15626841033897, −11.51215338994822, −11.22113797098853, −10.51804703771297, −10.13589478591845, −9.574676502985514, −9.157013862690961, −8.685213438469747, −7.949860735370696, −7.582657362492929, −6.861661363032583, −6.477022587511350, −6.017660191554568, −5.317923882955673, −4.701694479333378, −4.336421354913599, −3.365345732040736, −2.890184901881839, −2.644462510671464, −1.568899194019320, −0.7870039817200973, 0,
0.7870039817200973, 1.568899194019320, 2.644462510671464, 2.890184901881839, 3.365345732040736, 4.336421354913599, 4.701694479333378, 5.317923882955673, 6.017660191554568, 6.477022587511350, 6.861661363032583, 7.582657362492929, 7.949860735370696, 8.685213438469747, 9.157013862690961, 9.574676502985514, 10.13589478591845, 10.51804703771297, 11.22113797098853, 11.51215338994822, 12.15626841033897, 12.75267439282122, 13.12792140803411, 13.47370638772228, 13.97038575903942
