| L(s) = 1 | + 3-s + 1.47·5-s + 7-s + 9-s − 4.58·11-s + 3.11·13-s + 1.47·15-s − 6.58·17-s + 6.81·19-s + 21-s − 23-s − 2.83·25-s + 27-s − 7.30·29-s − 1.64·31-s − 4.58·33-s + 1.47·35-s − 8.10·37-s + 3.11·39-s − 9.87·41-s − 1.11·43-s + 1.47·45-s − 10.2·47-s + 49-s − 6.58·51-s − 5.47·53-s − 6.75·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.658·5-s + 0.377·7-s + 0.333·9-s − 1.38·11-s + 0.863·13-s + 0.380·15-s − 1.59·17-s + 1.56·19-s + 0.218·21-s − 0.208·23-s − 0.566·25-s + 0.192·27-s − 1.35·29-s − 0.294·31-s − 0.798·33-s + 0.248·35-s − 1.33·37-s + 0.498·39-s − 1.54·41-s − 0.170·43-s + 0.219·45-s − 1.49·47-s + 0.142·49-s − 0.922·51-s − 0.751·53-s − 0.911·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + 8.10T + 37T^{2} \) |
| 41 | \( 1 + 9.87T + 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 - 6.96T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−7.69983836402075293965766816075, −6.85897995574540721311179824843, −6.14581331503310050383597523245, −5.24344766672783959241678497733, −4.92234037204368515572225867887, −3.72799314032337939568877598408, −3.13786577984137574191803727846, −2.09120734402972391108175203718, −1.63345868575511824481165799233, 0,
1.63345868575511824481165799233, 2.09120734402972391108175203718, 3.13786577984137574191803727846, 3.72799314032337939568877598408, 4.92234037204368515572225867887, 5.24344766672783959241678497733, 6.14581331503310050383597523245, 6.85897995574540721311179824843, 7.69983836402075293965766816075
