| L(s) = 1 | − 2-s + 4-s + 4.05·5-s + 2.05·7-s − 8-s − 4.05·10-s + 0.740·11-s + 3.44·13-s − 2.05·14-s + 16-s + 7.31·17-s − 4.91·19-s + 4.05·20-s − 0.740·22-s − 2.65·23-s + 11.4·25-s − 3.44·26-s + 2.05·28-s + 0.948·29-s − 5.65·31-s − 32-s − 7.31·34-s + 8.31·35-s − 1.18·37-s + 4.91·38-s − 4.05·40-s + 41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.81·5-s + 0.775·7-s − 0.353·8-s − 1.28·10-s + 0.223·11-s + 0.956·13-s − 0.548·14-s + 0.250·16-s + 1.77·17-s − 1.12·19-s + 0.905·20-s − 0.157·22-s − 0.553·23-s + 2.28·25-s − 0.676·26-s + 0.387·28-s + 0.176·29-s − 1.01·31-s − 0.176·32-s − 1.25·34-s + 1.40·35-s − 0.195·37-s + 0.797·38-s − 0.640·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.219164856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219164856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 - 4.05T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 - 0.740T + 11T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 17 | \( 1 - 7.31T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 - 0.948T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 43 | \( 1 - 1.31T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−9.056821413860671093066392262929, −8.477870331135297702991958553129, −7.61848467592575351513414038810, −6.66855942405576256535112617093, −5.78256430640053837135511770292, −5.55100355497214265591328992963, −4.18468886768300388012167518867, −2.89607370947664318474802863065, −1.83414797329564878778212609630, −1.25047667174166953674965707580,
1.25047667174166953674965707580, 1.83414797329564878778212609630, 2.89607370947664318474802863065, 4.18468886768300388012167518867, 5.55100355497214265591328992963, 5.78256430640053837135511770292, 6.66855942405576256535112617093, 7.61848467592575351513414038810, 8.477870331135297702991958553129, 9.056821413860671093066392262929
