L(s) = 1 | + (−0.978 − 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (−2.71 − 3.02i)5-s + (0.309 − 0.951i)6-s + (−1.98 + 1.75i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (2.03 + 3.51i)10-s + (2.84 + 1.70i)11-s + (−0.5 + 0.866i)12-s + (0.914 + 2.81i)13-s + (2.30 − 1.30i)14-s + (3.28 − 2.38i)15-s + (0.669 + 0.743i)16-s + (4.37 − 0.929i)17-s + ⋯ |
L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.0603 + 0.574i)3-s + (0.456 + 0.203i)4-s + (−1.21 − 1.35i)5-s + (0.126 − 0.388i)6-s + (−0.749 + 0.661i)7-s + (−0.286 − 0.207i)8-s + (−0.326 − 0.0693i)9-s + (0.642 + 1.11i)10-s + (0.857 + 0.514i)11-s + (−0.144 + 0.250i)12-s + (0.253 + 0.780i)13-s + (0.615 − 0.347i)14-s + (0.848 − 0.616i)15-s + (0.167 + 0.185i)16-s + (1.06 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.770122 + 0.0773422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.770122 + 0.0773422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 7 | \( 1 + (1.98 - 1.75i)T \) |
| 11 | \( 1 + (-2.84 - 1.70i)T \) |
good | 5 | \( 1 + (2.71 + 3.02i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-0.914 - 2.81i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.37 + 0.929i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-6.68 + 2.97i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.81 + 3.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.324 - 0.235i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.09 + 4.54i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.904 - 8.60i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-5.50 - 3.99i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 + (0.119 - 0.0530i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (0.0143 - 0.0159i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (2.86 + 1.27i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (5.14 + 5.71i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.73 - 4.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.19 - 12.9i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.02 - 1.34i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-16.3 - 3.46i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.23 + 6.88i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.223 - 0.386i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.58 - 17.1i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.32024159596013908190507532693, −9.697034736137772112071841234169, −9.403104311317226886433194283441, −8.589077984676829376281879670624, −7.70729333693645860394539639219, −6.56654010076359122048979950710, −5.19604389848409196627361270052, −4.22269319550406653378471384054, −3.13012913885275154650534079417, −0.997235919315350598201205147787,
0.875079146648410727803461203673, 3.16006823056623655330001510809, 3.59986827985936951430386469100, 5.81310935457637412725545553537, 6.67115100158256351437013970428, 7.57226702588374944727871051564, 7.81684865584065223814089624000, 9.236292830351260579357928126034, 10.30865727872403957807190727804, 10.89659726253768602738649150382