L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 3·8-s + 9-s − 10-s − 2·11-s − 12-s + 7·13-s + 15-s − 16-s − 3·17-s − 18-s − 8·19-s − 20-s + 2·22-s + 23-s + 3·24-s − 4·25-s − 7·26-s + 27-s − 8·29-s − 30-s − 6·31-s − 5·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.94·13-s + 0.258·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.426·22-s + 0.208·23-s + 0.612·24-s − 4/5·25-s − 1.37·26-s + 0.192·27-s − 1.48·29-s − 0.182·30-s − 1.07·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.447776277741753717965000953864, −7.85089603938097020049763075249, −6.85225910550255752422765587213, −6.06993683661572108662828229954, −5.21772062058557241138792965365, −4.12261336825519308949399721062, −3.66657469151957635808336415666, −2.22233982841131204720440687781, −1.51621576440597365408738138313, 0,
1.51621576440597365408738138313, 2.22233982841131204720440687781, 3.66657469151957635808336415666, 4.12261336825519308949399721062, 5.21772062058557241138792965365, 6.06993683661572108662828229954, 6.85225910550255752422765587213, 7.85089603938097020049763075249, 8.447776277741753717965000953864