| L(s) = 1 | − 2.49·2-s − 3·3-s − 1.77·4-s + 21.0·5-s + 7.48·6-s − 29.8·7-s + 24.3·8-s + 9·9-s − 52.4·10-s − 11·11-s + 5.32·12-s − 36.4·13-s + 74.4·14-s − 63.0·15-s − 46.6·16-s + 28.9·17-s − 22.4·18-s − 33.1·19-s − 37.2·20-s + 89.5·21-s + 27.4·22-s − 8.54·23-s − 73.1·24-s + 316.·25-s + 90.9·26-s − 27·27-s + 52.9·28-s + ⋯ |
| L(s) = 1 | − 0.882·2-s − 0.577·3-s − 0.221·4-s + 1.88·5-s + 0.509·6-s − 1.61·7-s + 1.07·8-s + 0.333·9-s − 1.65·10-s − 0.301·11-s + 0.128·12-s − 0.777·13-s + 1.42·14-s − 1.08·15-s − 0.729·16-s + 0.412·17-s − 0.294·18-s − 0.400·19-s − 0.416·20-s + 0.930·21-s + 0.265·22-s − 0.0774·23-s − 0.622·24-s + 2.53·25-s + 0.686·26-s − 0.192·27-s + 0.357·28-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 | \( 1 + 2.49T + 8T^{2} \) |
| 5 | \( 1 - 21.0T + 125T^{2} \) |
| 7 | \( 1 + 29.8T + 343T^{2} \) |
| 13 | \( 1 + 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.54T + 1.21e4T^{2} \) |
| 29 | \( 1 - 260.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 171.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 21.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 194.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 250.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 46.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 679.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 59.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 346.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 728.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 219.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 861.T + 9.12e5T^{2} \) |
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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
−8.769325152626747142612389834280, −7.59916643278654148459718176741, −6.66919154930695374372139816287, −6.18272076445145613795147353789, −5.36087367431919834194445817530, −4.56093938244082001781595621086, −3.08687393262814357379121838314, −2.14621897689456580769428351516, −1.02671053157779785633699015613, 0,
1.02671053157779785633699015613, 2.14621897689456580769428351516, 3.08687393262814357379121838314, 4.56093938244082001781595621086, 5.36087367431919834194445817530, 6.18272076445145613795147353789, 6.66919154930695374372139816287, 7.59916643278654148459718176741, 8.769325152626747142612389834280
