| L(s) = 1 | − 5-s − 4·11-s − 2·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s − 2·37-s − 6·41-s + 4·43-s − 6·53-s + 4·55-s + 8·59-s − 2·61-s + 2·65-s + 4·67-s − 4·71-s + 10·73-s − 12·79-s + 4·83-s + 2·85-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.824·53-s + 0.539·55-s + 1.04·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.474·71-s + 1.17·73-s − 1.35·79-s + 0.439·83-s + 0.216·85-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.60212633057630, −13.19103295056154, −12.64309167437797, −12.18186162215575, −11.83812173016176, −11.20574423732670, −10.77433692309108, −10.24331363905652, −9.925343286492636, −9.338279863639238, −8.622670910917134, −8.265103735814435, −7.870157154949755, −7.312837565399262, −6.687687160506753, −6.417581426779719, −5.488845221786936, −5.191364263612259, −4.543008750596482, −4.193681861904724, −3.347221142637360, −2.752424281361998, −2.438606688135235, −1.554658507437359, −0.7025641917200781, 0,
0.7025641917200781, 1.554658507437359, 2.438606688135235, 2.752424281361998, 3.347221142637360, 4.193681861904724, 4.543008750596482, 5.191364263612259, 5.488845221786936, 6.417581426779719, 6.687687160506753, 7.312837565399262, 7.870157154949755, 8.265103735814435, 8.622670910917134, 9.338279863639238, 9.925343286492636, 10.24331363905652, 10.77433692309108, 11.20574423732670, 11.83812173016176, 12.18186162215575, 12.64309167437797, 13.19103295056154, 13.60212633057630
