| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 7-s + 3·8-s + 9-s + 10-s − 12-s − 2·13-s + 14-s − 15-s − 16-s − 17-s − 18-s + 20-s − 21-s − 2·23-s + 3·24-s + 25-s + 2·26-s + 27-s + 28-s − 6·29-s + 30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s − 0.417·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05776735801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05776735801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.08969314023764, −12.57766358091878, −12.10724537290883, −11.43191106807779, −11.20142697201196, −10.32987599006167, −10.10405813746370, −9.632907822484545, −9.339807940042545, −8.599000604986786, −8.272039743541725, −8.095985620008106, −7.195874431809104, −7.108026027024621, −6.480537614011273, −5.640432482661343, −5.212438813180929, −4.556325331844855, −4.058364934603766, −3.743898714031713, −2.882834255961030, −2.525099896227389, −1.608564962152313, −1.185030902606746, −0.07747614215011496,
0.07747614215011496, 1.185030902606746, 1.608564962152313, 2.525099896227389, 2.882834255961030, 3.743898714031713, 4.058364934603766, 4.556325331844855, 5.212438813180929, 5.640432482661343, 6.480537614011273, 7.108026027024621, 7.195874431809104, 8.095985620008106, 8.272039743541725, 8.599000604986786, 9.339807940042545, 9.632907822484545, 10.10405813746370, 10.32987599006167, 11.20142697201196, 11.43191106807779, 12.10724537290883, 12.57766358091878, 13.08969314023764
