L(s) = 1 | + 2.51·2-s + 3-s + 4.32·4-s − 4.31·5-s + 2.51·6-s + 5.85·8-s + 9-s − 10.8·10-s + 1.72·11-s + 4.32·12-s + 2.88·13-s − 4.31·15-s + 6.06·16-s + 4.93·17-s + 2.51·18-s − 2.44·19-s − 18.6·20-s + 4.34·22-s + 23-s + 5.85·24-s + 13.6·25-s + 7.26·26-s + 27-s + 8.05·29-s − 10.8·30-s − 0.915·31-s + 3.55·32-s + ⋯ |
L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.16·4-s − 1.93·5-s + 1.02·6-s + 2.06·8-s + 0.333·9-s − 3.43·10-s + 0.521·11-s + 1.24·12-s + 0.801·13-s − 1.11·15-s + 1.51·16-s + 1.19·17-s + 0.592·18-s − 0.560·19-s − 4.17·20-s + 0.927·22-s + 0.208·23-s + 1.19·24-s + 2.72·25-s + 1.42·26-s + 0.192·27-s + 1.49·29-s − 1.98·30-s − 0.164·31-s + 0.627·32-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
\(L(1)\) |
\(\approx\) |
\(5.621032486\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.621032486\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 4.31T + 5T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 + 0.915T + 31T^{2} \) |
| 37 | \( 1 + 9.15T + 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.399T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 - 1.83T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 9.00T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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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
−8.364934077702661964547387698664, −7.65905284829727731584245339384, −7.05240070733251801090213623565, −6.34193073208307064503812120023, −5.34822570047927979808779274032, −4.41599279571030381900391959887, −3.96038262482324170409092403081, −3.39157942877545765005571480216, −2.68161540090911009798529812564, −1.13098758380533523393978876197,
1.13098758380533523393978876197, 2.68161540090911009798529812564, 3.39157942877545765005571480216, 3.96038262482324170409092403081, 4.41599279571030381900391959887, 5.34822570047927979808779274032, 6.34193073208307064503812120023, 7.05240070733251801090213623565, 7.65905284829727731584245339384, 8.364934077702661964547387698664