L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.499 − 0.866i)4-s + (1.22 + 0.707i)11-s + (0.499 + 0.866i)16-s − 2·22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + 1.41i·29-s + (−1.22 − 0.707i)32-s + (1.22 − 0.707i)44-s + (−0.999 + 1.73i)46-s − 1.41i·50-s + (−1.22 − 0.707i)53-s + (−1.00 − 1.73i)58-s + 0.999·64-s + ⋯ |
L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.499 − 0.866i)4-s + (1.22 + 0.707i)11-s + (0.499 + 0.866i)16-s − 2·22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + 1.41i·29-s + (−1.22 − 0.707i)32-s + (1.22 − 0.707i)44-s + (−0.999 + 1.73i)46-s − 1.41i·50-s + (−1.22 − 0.707i)53-s + (−1.00 − 1.73i)58-s + 0.999·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4946526962\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4946526962\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.21850204661093033399877615259, −10.34348904407705027774282178700, −9.303193161721400397952824269371, −8.991725428697495458276048686635, −7.84273445823638068332700752822, −6.98762824746807386839413737328, −6.36194329576796973093361987275, −4.91076093647297368638248374155, −3.53857529263476151221789652631, −1.51270748961077959396563896666,
1.22447069934551854485032977313, 2.71155808384978503277722180903, 4.03905757834259701265667766827, 5.58395740293661999262320915742, 6.73258232742497397747311064116, 7.889531821328608948275817454481, 8.697862528249486234071116720372, 9.437568863336295526043237869777, 10.13444151591728093797579546615, 11.30002800260642342599704048754