L(s) = 1 | − 0.311·2-s − 3-s − 1.90·4-s + 0.311·6-s + 0.903·7-s + 1.21·8-s + 9-s + 11-s + 1.90·12-s − 2.90·13-s − 0.280·14-s + 3.42·16-s − 2.28·17-s − 0.311·18-s + 2.42·19-s − 0.903·21-s − 0.311·22-s − 4·23-s − 1.21·24-s + 0.903·26-s − 27-s − 1.71·28-s + 7.05·29-s − 2.62·31-s − 3.49·32-s − 33-s + 0.709·34-s + ⋯ |
L(s) = 1 | − 0.219·2-s − 0.577·3-s − 0.951·4-s + 0.127·6-s + 0.341·7-s + 0.429·8-s + 0.333·9-s + 0.301·11-s + 0.549·12-s − 0.805·13-s − 0.0750·14-s + 0.857·16-s − 0.553·17-s − 0.0733·18-s + 0.557·19-s − 0.197·21-s − 0.0663·22-s − 0.834·23-s − 0.247·24-s + 0.177·26-s − 0.192·27-s − 0.324·28-s + 1.30·29-s − 0.470·31-s − 0.617·32-s − 0.174·33-s + 0.121·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 - 0.903T + 7T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 0.949T + 47T^{2} \) |
| 53 | \( 1 + 0.815T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 + 7.24T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 - 6.13T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.955880833385090652789352521878, −8.958755080134150296825081630491, −8.214905430638894958424429450593, −7.27222957457784295880375963242, −6.28561739412942160519080176542, −5.09996759719112183530139638036, −4.64017506545693089220384394903, −3.41778385349178823269438616811, −1.63690961981755026140006473248, 0,
1.63690961981755026140006473248, 3.41778385349178823269438616811, 4.64017506545693089220384394903, 5.09996759719112183530139638036, 6.28561739412942160519080176542, 7.27222957457784295880375963242, 8.214905430638894958424429450593, 8.958755080134150296825081630491, 9.955880833385090652789352521878