L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 2·11-s + 6·13-s + 2·14-s + 16-s − 2·17-s − 2·22-s + 4·23-s + 6·26-s + 2·28-s − 8·31-s + 32-s − 2·34-s + 2·37-s − 2·41-s − 4·43-s − 2·44-s + 4·46-s + 8·47-s − 3·49-s + 6·52-s − 6·53-s + 2·56-s − 10·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.426·22-s + 0.834·23-s + 1.17·26-s + 0.377·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.301·44-s + 0.589·46-s + 1.16·47-s − 3/7·49-s + 0.832·52-s − 0.824·53-s + 0.267·56-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.298486058\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.298486058\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−10.99462797763282172519820421382, −10.70113359430777348682650149452, −9.170289817714090369137184862208, −8.300870184686946312404901208144, −7.34846116247576004745703536007, −6.23125073665726777261133626481, −5.32042888975375281428118939697, −4.30565082727211113228432978844, −3.15570185065497901164785729690, −1.63197914976626677836987834740,
1.63197914976626677836987834740, 3.15570185065497901164785729690, 4.30565082727211113228432978844, 5.32042888975375281428118939697, 6.23125073665726777261133626481, 7.34846116247576004745703536007, 8.300870184686946312404901208144, 9.170289817714090369137184862208, 10.70113359430777348682650149452, 10.99462797763282172519820421382