| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 13-s + 16-s − 6·17-s + 4·19-s − 4·22-s + 8·23-s − 26-s − 6·29-s − 8·31-s + 32-s − 6·34-s + 10·37-s + 4·38-s + 6·41-s − 4·43-s − 4·44-s + 8·46-s − 7·49-s − 52-s − 10·53-s − 6·58-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.852·22-s + 1.66·23-s − 0.196·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s − 49-s − 0.138·52-s − 1.37·53-s − 0.787·58-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.46033849916812203391098288116, −7.17431492168162127839214453508, −6.19945663799169026558698300536, −5.46345834037435546276923044570, −4.88813975335240649367046900676, −4.19696708190826095524664922309, −3.11832356061336026608331109958, −2.60016702005141062593276209503, −1.54138230616847407701907689447, 0,
1.54138230616847407701907689447, 2.60016702005141062593276209503, 3.11832356061336026608331109958, 4.19696708190826095524664922309, 4.88813975335240649367046900676, 5.46345834037435546276923044570, 6.19945663799169026558698300536, 7.17431492168162127839214453508, 7.46033849916812203391098288116
