| L(s) = 1 | + 3-s − 4.37·5-s + 7-s + 9-s − 4·11-s − 4.37·13-s − 4.37·15-s + 6.74·17-s − 4·19-s + 21-s + 23-s + 14.1·25-s + 27-s + 3.62·29-s + 4.74·31-s − 4·33-s − 4.37·35-s + 8.37·37-s − 4.37·39-s + 9.11·41-s + 11.1·43-s − 4.37·45-s − 10.3·47-s + 49-s + 6.74·51-s − 6.74·53-s + 17.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.95·5-s + 0.377·7-s + 0.333·9-s − 1.20·11-s − 1.21·13-s − 1.12·15-s + 1.63·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s + 2.82·25-s + 0.192·27-s + 0.673·29-s + 0.852·31-s − 0.696·33-s − 0.739·35-s + 1.37·37-s − 0.700·39-s + 1.42·41-s + 1.69·43-s − 0.651·45-s − 1.51·47-s + 0.142·49-s + 0.944·51-s − 0.926·53-s + 2.35·55-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 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 6.74T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−7.64971709939500364914916256556, −7.30305426305832870301351715178, −6.21101894566425267893421697640, −5.12342173864766875080406065540, −4.54168289208179052538929075147, −4.02809502772808730327913048948, −2.87981364504074882703478227046, −2.73644350892143027180639791080, −1.08709967068228190391902878698, 0,
1.08709967068228190391902878698, 2.73644350892143027180639791080, 2.87981364504074882703478227046, 4.02809502772808730327913048948, 4.54168289208179052538929075147, 5.12342173864766875080406065540, 6.21101894566425267893421697640, 7.30305426305832870301351715178, 7.64971709939500364914916256556
