L(s) = 1 | − 2.33·2-s + 3-s + 3.47·4-s − 0.253·5-s − 2.33·6-s − 1.85·7-s − 3.44·8-s + 9-s + 0.592·10-s + 5.63·11-s + 3.47·12-s + 3.74·13-s + 4.34·14-s − 0.253·15-s + 1.12·16-s − 17-s − 2.33·18-s − 0.451·19-s − 0.879·20-s − 1.85·21-s − 13.1·22-s − 5.63·23-s − 3.44·24-s − 4.93·25-s − 8.77·26-s + 27-s − 6.45·28-s + ⋯ |
L(s) = 1 | − 1.65·2-s + 0.577·3-s + 1.73·4-s − 0.113·5-s − 0.955·6-s − 0.702·7-s − 1.21·8-s + 0.333·9-s + 0.187·10-s + 1.69·11-s + 1.00·12-s + 1.03·13-s + 1.16·14-s − 0.0653·15-s + 0.280·16-s − 0.242·17-s − 0.551·18-s − 0.103·19-s − 0.196·20-s − 0.405·21-s − 2.80·22-s − 1.17·23-s − 0.704·24-s − 0.987·25-s − 1.72·26-s + 0.192·27-s − 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.088596233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.088596233\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 + 0.253T + 5T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 19 | \( 1 + 0.451T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 - 6.64T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 2.91T + 41T^{2} \) |
| 43 | \( 1 + 0.243T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 + 0.372T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 + 4.60T + 73T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.440507198707692743849866309081, −8.090567718164490224506437756532, −7.14678718996875790586611226188, −6.40900831068849108142012603108, −6.10935215329548023807857031432, −4.31424719453939088757314293949, −3.73419136174183633736564260008, −2.64493752778490188166706491204, −1.65648421389055333053514730831, −0.78517614615530794767801419658,
0.78517614615530794767801419658, 1.65648421389055333053514730831, 2.64493752778490188166706491204, 3.73419136174183633736564260008, 4.31424719453939088757314293949, 6.10935215329548023807857031432, 6.40900831068849108142012603108, 7.14678718996875790586611226188, 8.090567718164490224506437756532, 8.440507198707692743849866309081