| L(s) = 1 | − 0.121·2-s + 3·3-s − 7.98·4-s + 12.2·5-s − 0.363·6-s + 1.74·7-s + 1.93·8-s + 9·9-s − 1.48·10-s + 11·11-s − 23.9·12-s − 16.8·13-s − 0.210·14-s + 36.8·15-s + 63.6·16-s − 69.8·17-s − 1.08·18-s + 76.4·19-s − 98.0·20-s + 5.22·21-s − 1.33·22-s − 164.·23-s + 5.80·24-s + 25.8·25-s + 2.04·26-s + 27·27-s − 13.9·28-s + ⋯ |
| L(s) = 1 | − 0.0427·2-s + 0.577·3-s − 0.998·4-s + 1.09·5-s − 0.0247·6-s + 0.0940·7-s + 0.0854·8-s + 0.333·9-s − 0.0470·10-s + 0.301·11-s − 0.576·12-s − 0.359·13-s − 0.00402·14-s + 0.634·15-s + 0.994·16-s − 0.996·17-s − 0.0142·18-s + 0.923·19-s − 1.09·20-s + 0.0543·21-s − 0.0129·22-s − 1.48·23-s + 0.0493·24-s + 0.206·25-s + 0.0153·26-s + 0.192·27-s − 0.0939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 0.121T + 8T^{2} \) |
| 5 | \( 1 - 12.2T + 125T^{2} \) |
| 7 | \( 1 - 1.74T + 343T^{2} \) |
| 13 | \( 1 + 16.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 69.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 332.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 356.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 422.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 75.2T + 2.05e5T^{2} \) |
| 67 | \( 1 - 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 202.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 322.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 25.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 309.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 53.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 232.T + 9.12e5T^{2} \) |
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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
−8.462020083742290173103740389508, −7.893004306429166919194306509717, −6.80388479704819312665677079361, −5.94430686797408736385110017982, −5.13409548792426128269733816580, −4.32312422846139434374376743375, −3.42186374055998968078449473980, −2.26468324907869753409985327970, −1.41424997742856864882106526420, 0,
1.41424997742856864882106526420, 2.26468324907869753409985327970, 3.42186374055998968078449473980, 4.32312422846139434374376743375, 5.13409548792426128269733816580, 5.94430686797408736385110017982, 6.80388479704819312665677079361, 7.893004306429166919194306509717, 8.462020083742290173103740389508
