| L(s) = 1 | + 3-s − 0.434·5-s − 2.25·7-s + 9-s + 3.49·11-s + 5.02·13-s − 0.434·15-s − 0.922·17-s + 5.87·19-s − 2.25·21-s + 23-s − 4.81·25-s + 27-s − 29-s + 9.57·31-s + 3.49·33-s + 0.980·35-s + 2.89·37-s + 5.02·39-s − 5.70·41-s + 4.57·43-s − 0.434·45-s − 1.28·47-s − 1.89·49-s − 0.922·51-s − 9.88·53-s − 1.51·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.194·5-s − 0.854·7-s + 0.333·9-s + 1.05·11-s + 1.39·13-s − 0.112·15-s − 0.223·17-s + 1.34·19-s − 0.493·21-s + 0.208·23-s − 0.962·25-s + 0.192·27-s − 0.185·29-s + 1.71·31-s + 0.607·33-s + 0.165·35-s + 0.475·37-s + 0.804·39-s − 0.890·41-s + 0.697·43-s − 0.0646·45-s − 0.187·47-s − 0.270·49-s − 0.129·51-s − 1.35·53-s − 0.204·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.725743429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725743429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 0.434T + 5T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 0.922T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + 9.88T + 53T^{2} \) |
| 59 | \( 1 + 0.268T + 59T^{2} \) |
| 61 | \( 1 + 4.59T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 2.10T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 - 0.502T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.899852322923072769387802964487, −7.15513933908983826512302344815, −6.33110379637867745402535283108, −6.09243850541430237025173251746, −4.92871841899464999249192893905, −4.04790523188962740822310065469, −3.47921519907534649883109069844, −2.92213407385739563599553695712, −1.69446228610328820895965450073, −0.840156139820717569782745286461,
0.840156139820717569782745286461, 1.69446228610328820895965450073, 2.92213407385739563599553695712, 3.47921519907534649883109069844, 4.04790523188962740822310065469, 4.92871841899464999249192893905, 6.09243850541430237025173251746, 6.33110379637867745402535283108, 7.15513933908983826512302344815, 7.899852322923072769387802964487
