L(s) = 1 | − 1.34·2-s − 2.87·3-s − 0.184·4-s + 3.87·6-s − 0.347·7-s + 2.94·8-s + 5.29·9-s − 2.22·11-s + 0.532·12-s + 2.57·13-s + 0.467·14-s − 3.59·16-s − 0.467·17-s − 7.12·18-s + 21-s + 3·22-s + 2.69·23-s − 8.47·24-s − 3.46·26-s − 6.59·27-s + 0.0641·28-s + 6.87·29-s + 7.10·31-s − 1.04·32-s + 6.41·33-s + 0.630·34-s − 0.977·36-s + ⋯ |
L(s) = 1 | − 0.952·2-s − 1.66·3-s − 0.0923·4-s + 1.58·6-s − 0.131·7-s + 1.04·8-s + 1.76·9-s − 0.671·11-s + 0.153·12-s + 0.713·13-s + 0.125·14-s − 0.899·16-s − 0.113·17-s − 1.68·18-s + 0.218·21-s + 0.639·22-s + 0.561·23-s − 1.73·24-s − 0.680·26-s − 1.26·27-s + 0.0121·28-s + 1.27·29-s + 1.27·31-s − 0.184·32-s + 1.11·33-s + 0.108·34-s − 0.162·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5859592122\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5859592122\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 7 | \( 1 + 0.347T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 - 6.30T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 + 1.38T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 9.45T + 97T^{2} \) |
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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
−7.925520007989315814113230675667, −6.82183911011556705288425729536, −6.57723296998921608867352947548, −5.70286344696898770416583917933, −4.96010219377992190620177576792, −4.59294923691610902385761941072, −3.59219918406655310276706902359, −2.31980406943432815231010789100, −1.08620044634322925894159128700, −0.60325252527394366487365240037,
0.60325252527394366487365240037, 1.08620044634322925894159128700, 2.31980406943432815231010789100, 3.59219918406655310276706902359, 4.59294923691610902385761941072, 4.96010219377992190620177576792, 5.70286344696898770416583917933, 6.57723296998921608867352947548, 6.82183911011556705288425729536, 7.925520007989315814113230675667