| L(s) = 1 | + 2-s + 0.732·3-s + 4-s + 2.73·5-s + 0.732·6-s + 7-s + 8-s − 2.46·9-s + 2.73·10-s − 3.46·11-s + 0.732·12-s − 1.46·13-s + 14-s + 2·15-s + 16-s − 0.732·17-s − 2.46·18-s − 2·19-s + 2.73·20-s + 0.732·21-s − 3.46·22-s + 23-s + 0.732·24-s + 2.46·25-s − 1.46·26-s − 4·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s + 1.22·5-s + 0.298·6-s + 0.377·7-s + 0.353·8-s − 0.821·9-s + 0.863·10-s − 1.04·11-s + 0.211·12-s − 0.406·13-s + 0.267·14-s + 0.516·15-s + 0.250·16-s − 0.177·17-s − 0.580·18-s − 0.458·19-s + 0.610·20-s + 0.159·21-s − 0.738·22-s + 0.208·23-s + 0.149·24-s + 0.492·25-s − 0.287·26-s − 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.440218467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.440218467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 - 0.535T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 0.928T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.73T + 97T^{2} \) |
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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
−11.70067506929354483704720725075, −10.67695257272027306870633664533, −9.910464446373822919909083333012, −8.755371363738823579525397429705, −7.84429094861647662196212062621, −6.51333532832681734755701038830, −5.58202829084117856042817900705, −4.74557933607194402922955567371, −2.99236055587300241757013965243, −2.11107970094479299698612823224,
2.11107970094479299698612823224, 2.99236055587300241757013965243, 4.74557933607194402922955567371, 5.58202829084117856042817900705, 6.51333532832681734755701038830, 7.84429094861647662196212062621, 8.755371363738823579525397429705, 9.910464446373822919909083333012, 10.67695257272027306870633664533, 11.70067506929354483704720725075
