| L(s) = 1 | + 1.56·2-s + 0.438·4-s + 7-s − 2.43·8-s − 2.56·11-s − 4.56·13-s + 1.56·14-s − 4.68·16-s − 4.56·17-s + 1.12·19-s − 4·22-s − 5.12·23-s − 7.12·26-s + 0.438·28-s + 5.68·29-s − 2.43·32-s − 7.12·34-s − 6·37-s + 1.75·38-s + 3.12·41-s − 9.12·43-s − 1.12·44-s − 8·46-s + 3.68·47-s + 49-s − 1.99·52-s + 3.12·53-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.219·4-s + 0.377·7-s − 0.862·8-s − 0.772·11-s − 1.26·13-s + 0.417·14-s − 1.17·16-s − 1.10·17-s + 0.257·19-s − 0.852·22-s − 1.06·23-s − 1.39·26-s + 0.0828·28-s + 1.05·29-s − 0.431·32-s − 1.22·34-s − 0.986·37-s + 0.284·38-s + 0.487·41-s − 1.39·43-s − 0.169·44-s − 1.17·46-s + 0.537·47-s + 0.142·49-s − 0.277·52-s + 0.428·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.974461626590085730877765775247, −8.214377903058339903323868904272, −7.27613823576566809276337895181, −6.42724403992767984401364677976, −5.46027936825311731931033431195, −4.82960689747777365542531811520, −4.16707225803529488355952294427, −2.97980200135367741907991651169, −2.15207966297178687611243312945, 0,
2.15207966297178687611243312945, 2.97980200135367741907991651169, 4.16707225803529488355952294427, 4.82960689747777365542531811520, 5.46027936825311731931033431195, 6.42724403992767984401364677976, 7.27613823576566809276337895181, 8.214377903058339903323868904272, 8.974461626590085730877765775247
