L(s) = 1 | − 2-s − 3-s − 4-s + 3·5-s + 6-s + 3·8-s + 9-s − 3·10-s − 2·11-s + 12-s − 5·13-s − 3·15-s − 16-s + 2·17-s − 18-s + 2·19-s − 3·20-s + 2·22-s − 3·23-s − 3·24-s + 4·25-s + 5·26-s − 27-s + 29-s + 3·30-s + 8·31-s − 5·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 1.34·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.948·10-s − 0.603·11-s + 0.288·12-s − 1.38·13-s − 0.774·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.458·19-s − 0.670·20-s + 0.426·22-s − 0.625·23-s − 0.612·24-s + 4/5·25-s + 0.980·26-s − 0.192·27-s + 0.185·29-s + 0.547·30-s + 1.43·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.77764078297471693556322200806, −7.12387920269612571731567027372, −6.24871047563146648441610633636, −5.51959138254797986596774227822, −4.98953158489082943721761946277, −4.35151866125273887616739258065, −2.97981337164490558404793578016, −2.08449186556912797626020008638, −1.17639925236920594937176961360, 0,
1.17639925236920594937176961360, 2.08449186556912797626020008638, 2.97981337164490558404793578016, 4.35151866125273887616739258065, 4.98953158489082943721761946277, 5.51959138254797986596774227822, 6.24871047563146648441610633636, 7.12387920269612571731567027372, 7.77764078297471693556322200806