| L(s) = 1 | + 2-s + 4-s + 3.55·5-s − 0.319·7-s + 8-s + 3.55·10-s + 4.45·11-s − 4.67·13-s − 0.319·14-s + 16-s + 1.13·17-s + 1.85·19-s + 3.55·20-s + 4.45·22-s + 0.144·23-s + 7.62·25-s − 4.67·26-s − 0.319·28-s − 4.35·29-s + 0.671·31-s + 32-s + 1.13·34-s − 1.13·35-s + 7.58·37-s + 1.85·38-s + 3.55·40-s + 2.17·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.58·5-s − 0.120·7-s + 0.353·8-s + 1.12·10-s + 1.34·11-s − 1.29·13-s − 0.0853·14-s + 0.250·16-s + 0.275·17-s + 0.426·19-s + 0.794·20-s + 0.949·22-s + 0.0300·23-s + 1.52·25-s − 0.917·26-s − 0.0603·28-s − 0.809·29-s + 0.120·31-s + 0.176·32-s + 0.194·34-s − 0.191·35-s + 1.24·37-s + 0.301·38-s + 0.561·40-s + 0.340·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.571852711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.571852711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 + 0.319T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 - 0.144T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - 0.671T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 0.805T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 8.09T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 - 5.05T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−9.778582163000875754932336917963, −8.971496014559583999567074482679, −7.71217489796338799404183521777, −6.75369308564766267990941057973, −6.21205416105970764410366038150, −5.36880537257606554563441606635, −4.63445027904857523711010810349, −3.41585108268184474559021629094, −2.36824955360891924686353550173, −1.44752471834705312279264101158,
1.44752471834705312279264101158, 2.36824955360891924686353550173, 3.41585108268184474559021629094, 4.63445027904857523711010810349, 5.36880537257606554563441606635, 6.21205416105970764410366038150, 6.75369308564766267990941057973, 7.71217489796338799404183521777, 8.971496014559583999567074482679, 9.778582163000875754932336917963
