L(s) = 1 | − 2-s + 4-s + 0.0737·5-s + 2.82·7-s − 8-s − 0.0737·10-s − 0.846·11-s + 6.61·13-s − 2.82·14-s + 16-s + 2.99·17-s − 4.47·19-s + 0.0737·20-s + 0.846·22-s − 3.02·23-s − 4.99·25-s − 6.61·26-s + 2.82·28-s − 7.80·29-s + 8.54·31-s − 32-s − 2.99·34-s + 0.207·35-s − 4.28·37-s + 4.47·38-s − 0.0737·40-s − 4.12·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.0329·5-s + 1.06·7-s − 0.353·8-s − 0.0233·10-s − 0.255·11-s + 1.83·13-s − 0.753·14-s + 0.250·16-s + 0.725·17-s − 1.02·19-s + 0.0164·20-s + 0.180·22-s − 0.630·23-s − 0.998·25-s − 1.29·26-s + 0.533·28-s − 1.44·29-s + 1.53·31-s − 0.176·32-s − 0.512·34-s + 0.0351·35-s − 0.703·37-s + 0.726·38-s − 0.0116·40-s − 0.643·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 0.0737T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 0.846T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 8.54T + 31T^{2} \) |
| 37 | \( 1 + 4.28T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 + 0.662T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.88T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 + 1.58T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.85887578422523421996822934428, −6.82277892217361755559756191357, −6.13153761345964988621852727029, −5.59700578947432199823175154591, −4.64444928772952023889911916977, −3.84122661880151192121371955506, −3.06669004344495658602315964435, −1.74942648528724910978481776221, −1.50194842079194512591799830725, 0,
1.50194842079194512591799830725, 1.74942648528724910978481776221, 3.06669004344495658602315964435, 3.84122661880151192121371955506, 4.64444928772952023889911916977, 5.59700578947432199823175154591, 6.13153761345964988621852727029, 6.82277892217361755559756191357, 7.85887578422523421996822934428