| L(s) = 1 | − 3.73·5-s + 6.46·13-s − 5.73·17-s + 8.92·25-s + 10.6·29-s − 9.39·37-s − 8·41-s − 7·49-s − 4·53-s + 15.3·61-s − 24.1·65-s − 16.8·73-s + 21.3·85-s + 0.660·89-s − 18·97-s − 20·101-s − 14.3·109-s + 4.12·113-s + ⋯ |
| L(s) = 1 | − 1.66·5-s + 1.79·13-s − 1.39·17-s + 1.78·25-s + 1.97·29-s − 1.54·37-s − 1.24·41-s − 49-s − 0.549·53-s + 1.97·61-s − 2.99·65-s − 1.97·73-s + 2.32·85-s + 0.0699·89-s − 1.82·97-s − 1.99·101-s − 1.37·109-s + 0.387·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 0.660T + 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−8.549664320534920130452322868605, −7.940983534529901900071170250520, −6.79763238598130107172210796561, −6.55323386207423441565482066239, −5.21342169130791343157784205613, −4.32102035360810672281463146472, −3.74822689864830892315566360089, −2.90937050589233368890153506241, −1.35113073394202297598788621155, 0,
1.35113073394202297598788621155, 2.90937050589233368890153506241, 3.74822689864830892315566360089, 4.32102035360810672281463146472, 5.21342169130791343157784205613, 6.55323386207423441565482066239, 6.79763238598130107172210796561, 7.940983534529901900071170250520, 8.549664320534920130452322868605
