| L(s) = 1 | + 3·3-s − 7-s + 6·9-s − 2·11-s + 13-s − 2·19-s − 3·21-s − 23-s + 9·27-s − 3·29-s + 31-s − 6·33-s + 2·37-s + 3·39-s − 41-s + 8·43-s − 5·47-s + 49-s + 6·53-s − 6·57-s + 6·61-s − 6·63-s − 10·67-s − 3·69-s − 7·71-s − 13·73-s + 2·77-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s − 0.603·11-s + 0.277·13-s − 0.458·19-s − 0.654·21-s − 0.208·23-s + 1.73·27-s − 0.557·29-s + 0.179·31-s − 1.04·33-s + 0.328·37-s + 0.480·39-s − 0.156·41-s + 1.21·43-s − 0.729·47-s + 1/7·49-s + 0.824·53-s − 0.794·57-s + 0.768·61-s − 0.755·63-s − 1.22·67-s − 0.361·69-s − 0.830·71-s − 1.52·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.54327051052805, −14.00722102160621, −13.44296279679381, −13.08157011919859, −12.82277441880770, −12.07578843670064, −11.50127631396709, −10.71237406270495, −10.32784879838931, −9.778182822682567, −9.306748068590970, −8.770122023638966, −8.416379722939402, −7.816667111643395, −7.391444305696002, −6.888732403259513, −6.120646194573544, −5.582159672454353, −4.703167917216563, −4.123741000176648, −3.671750461789798, −2.933025867805717, −2.586973208918777, −1.903979942644490, −1.178366086750267, 0,
1.178366086750267, 1.903979942644490, 2.586973208918777, 2.933025867805717, 3.671750461789798, 4.123741000176648, 4.703167917216563, 5.582159672454353, 6.120646194573544, 6.888732403259513, 7.391444305696002, 7.816667111643395, 8.416379722939402, 8.770122023638966, 9.306748068590970, 9.778182822682567, 10.32784879838931, 10.71237406270495, 11.50127631396709, 12.07578843670064, 12.82277441880770, 13.08157011919859, 13.44296279679381, 14.00722102160621, 14.54327051052805
