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 + 13-s − 3·15-s − 16-s − 17-s − 18-s − 3·20-s + 2·22-s + 23-s − 3·24-s + 4·25-s − 26-s − 27-s + 3·30-s − 10·31-s − 5·32-s + 2·33-s + 34-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 + 0.277·13-s − 0.774·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.670·20-s + 0.426·22-s + 0.208·23-s − 0.612·24-s + 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.547·30-s − 1.79·31-s − 0.883·32-s + 0.348·33-s + 0.171·34-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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.397434739538635308522811416265, −7.54953302210333067765232210194, −6.77457042774669984755651614642, −5.88307228368470036158662172078, −5.30403377076346159064185834454, −4.63122348669204324795694350251, −3.48853796874843375393901457772, −2.13980358358791365317810415210, −1.36116943915849509584993081936, 0,
1.36116943915849509584993081936, 2.13980358358791365317810415210, 3.48853796874843375393901457772, 4.63122348669204324795694350251, 5.30403377076346159064185834454, 5.88307228368470036158662172078, 6.77457042774669984755651614642, 7.54953302210333067765232210194, 8.397434739538635308522811416265