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
−10.89659726253768602738649150382, −10.30865727872403957807190727804, −9.236292830351260579357928126034, −7.81684865584065223814089624000, −7.57226702588374944727871051564, −6.67115100158256351437013970428, −5.81310935457637412725545553537, −3.59986827985936951430386469100, −3.16006823056623655330001510809, −0.875079146648410727803461203673,
0.997235919315350598201205147787, 3.13012913885275154650534079417, 4.22269319550406653378471384054, 5.19604389848409196627361270052, 6.56654010076359122048979950710, 7.70729333693645860394539639219, 8.589077984676829376281879670624, 9.403104311317226886433194283441, 9.697034736137772112071841234169, 11.32024159596013908190507532693