L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·11-s − 13-s + 14-s + 16-s + 6·17-s − 6·19-s − 20-s − 2·22-s − 23-s − 4·25-s + 26-s − 28-s + 5·29-s − 8·31-s − 32-s − 6·34-s + 35-s + 3·37-s + 6·38-s + 40-s + 9·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 0.928·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.493·37-s + 0.973·38-s + 0.158·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.182202421062221226969372052166, −7.902170262885049680261636447786, −6.94545470472771959839238122839, −6.27649313876450289167807265376, −5.48165904063050717242636743713, −4.27327951186921093207944603956, −3.54654475126023085816390171508, −2.50482061136141444833796675660, −1.34263154608521710049623609300, 0,
1.34263154608521710049623609300, 2.50482061136141444833796675660, 3.54654475126023085816390171508, 4.27327951186921093207944603956, 5.48165904063050717242636743713, 6.27649313876450289167807265376, 6.94545470472771959839238122839, 7.902170262885049680261636447786, 8.182202421062221226969372052166