| L(s) = 1 | + 1.67·2-s + 3.26·3-s + 0.821·4-s + 5-s + 5.48·6-s − 7-s − 1.97·8-s + 7.64·9-s + 1.67·10-s + 2.67·11-s + 2.67·12-s − 4.80·13-s − 1.67·14-s + 3.26·15-s − 4.96·16-s − 3.20·17-s + 12.8·18-s + 2.76·19-s + 0.821·20-s − 3.26·21-s + 4.50·22-s − 23-s − 6.46·24-s + 25-s − 8.06·26-s + 15.1·27-s − 0.821·28-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 1.88·3-s + 0.410·4-s + 0.447·5-s + 2.23·6-s − 0.377·7-s − 0.700·8-s + 2.54·9-s + 0.531·10-s + 0.807·11-s + 0.773·12-s − 1.33·13-s − 0.448·14-s + 0.842·15-s − 1.24·16-s − 0.777·17-s + 3.02·18-s + 0.633·19-s + 0.183·20-s − 0.712·21-s + 0.959·22-s − 0.208·23-s − 1.31·24-s + 0.200·25-s − 1.58·26-s + 2.91·27-s − 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.749242850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.749242850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 3 | \( 1 - 3.26T + 3T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 8.00T + 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 - 7.00T + 47T^{2} \) |
| 53 | \( 1 + 0.178T + 53T^{2} \) |
| 59 | \( 1 + 8.96T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 3.73T + 97T^{2} \) |
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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
−9.751556834599384882070343431355, −9.422603620275324136603241576376, −8.786766688066170281612969553976, −7.53985674992844710600210502727, −6.89890222106119575636263868743, −5.69065797399379242164433146151, −4.48364003195188766685276688995, −3.79683237544495915279186893480, −2.85711881460888311745952164264, −2.04003315769298212770123398841,
2.04003315769298212770123398841, 2.85711881460888311745952164264, 3.79683237544495915279186893480, 4.48364003195188766685276688995, 5.69065797399379242164433146151, 6.89890222106119575636263868743, 7.53985674992844710600210502727, 8.786766688066170281612969553976, 9.422603620275324136603241576376, 9.751556834599384882070343431355
