| L(s) = 1 | + 3.90·2-s + 3·3-s + 7.28·4-s + 0.418·5-s + 11.7·6-s − 20.3·7-s − 2.81·8-s + 9·9-s + 1.63·10-s + 11·11-s + 21.8·12-s + 21.0·13-s − 79.4·14-s + 1.25·15-s − 69.2·16-s + 6.74·17-s + 35.1·18-s + 102.·19-s + 3.04·20-s − 61.0·21-s + 42.9·22-s − 62.2·23-s − 8.43·24-s − 124.·25-s + 82.1·26-s + 27·27-s − 148.·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.577·3-s + 0.910·4-s + 0.0374·5-s + 0.797·6-s − 1.09·7-s − 0.124·8-s + 0.333·9-s + 0.0517·10-s + 0.301·11-s + 0.525·12-s + 0.448·13-s − 1.51·14-s + 0.0215·15-s − 1.08·16-s + 0.0961·17-s + 0.460·18-s + 1.24·19-s + 0.0340·20-s − 0.633·21-s + 0.416·22-s − 0.564·23-s − 0.0717·24-s − 0.998·25-s + 0.619·26-s + 0.192·27-s − 0.999·28-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 | \( 1 - 3.90T + 8T^{2} \) |
| 5 | \( 1 - 0.418T + 125T^{2} \) |
| 7 | \( 1 + 20.3T + 343T^{2} \) |
| 13 | \( 1 - 21.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.74T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 443.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 221.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 421.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 273.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 854.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 229.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 712.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−8.440244024052597956600382356446, −7.34539796932000503833322055532, −6.65237216712765701508895129375, −5.88723980027999058826798664554, −5.18100634540583697901238981133, −4.05878995574549645903726092871, −3.48103704948477713725745677190, −2.87838323161582742694795854884, −1.65526578860843181500317315420, 0,
1.65526578860843181500317315420, 2.87838323161582742694795854884, 3.48103704948477713725745677190, 4.05878995574549645903726092871, 5.18100634540583697901238981133, 5.88723980027999058826798664554, 6.65237216712765701508895129375, 7.34539796932000503833322055532, 8.440244024052597956600382356446
