L(s) = 1 | + 2.04·2-s + 3-s + 2.17·4-s − 1.94·5-s + 2.04·6-s + 0.358·8-s + 9-s − 3.97·10-s − 5.66·11-s + 2.17·12-s + 6.17·13-s − 1.94·15-s − 3.61·16-s − 7.23·17-s + 2.04·18-s − 0.790·19-s − 4.22·20-s − 11.5·22-s − 23-s + 0.358·24-s − 1.22·25-s + 12.6·26-s + 27-s − 5.34·29-s − 3.97·30-s − 0.911·31-s − 8.11·32-s + ⋯ |
L(s) = 1 | + 1.44·2-s + 0.577·3-s + 1.08·4-s − 0.869·5-s + 0.834·6-s + 0.126·8-s + 0.333·9-s − 1.25·10-s − 1.70·11-s + 0.628·12-s + 1.71·13-s − 0.501·15-s − 0.904·16-s − 1.75·17-s + 0.481·18-s − 0.181·19-s − 0.945·20-s − 2.46·22-s − 0.208·23-s + 0.0731·24-s − 0.244·25-s + 2.47·26-s + 0.192·27-s − 0.991·29-s − 0.725·30-s − 0.163·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 11 | \( 1 + 5.66T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 + 0.790T + 19T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 0.911T + 31T^{2} \) |
| 37 | \( 1 + 0.973T + 37T^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 + 8.75T + 71T^{2} \) |
| 73 | \( 1 - 0.875T + 73T^{2} \) |
| 79 | \( 1 - 5.96T + 79T^{2} \) |
| 83 | \( 1 - 0.273T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 3.09T + 97T^{2} \) |
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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
−8.293977027120489099982408312683, −7.39868559613912276682901511463, −6.61761679878243070006442285778, −5.80829617225615042312108605671, −5.04190490166781138887423913726, −4.14455557909847818309876853063, −3.74734794543129306197985519933, −2.84737039920162183785035346070, −2.01970497036682635172390964989, 0,
2.01970497036682635172390964989, 2.84737039920162183785035346070, 3.74734794543129306197985519933, 4.14455557909847818309876853063, 5.04190490166781138887423913726, 5.80829617225615042312108605671, 6.61761679878243070006442285778, 7.39868559613912276682901511463, 8.293977027120489099982408312683