| L(s) = 1 | + (−0.988 + 0.149i)2-s + (0.826 + 0.563i)3-s + (0.955 − 0.294i)4-s + (−0.0829 + 1.10i)5-s + (−0.900 − 0.433i)6-s + (−2.15 + 1.52i)7-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (−0.0829 − 1.10i)10-s + (−0.896 + 2.28i)11-s + (0.955 + 0.294i)12-s + (1.75 + 2.19i)13-s + (1.90 − 1.83i)14-s + (−0.692 + 0.867i)15-s + (0.826 − 0.563i)16-s + (−5.23 + 4.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 + 0.105i)2-s + (0.477 + 0.325i)3-s + (0.477 − 0.147i)4-s + (−0.0371 + 0.495i)5-s + (−0.367 − 0.177i)6-s + (−0.815 + 0.578i)7-s + (−0.318 + 0.153i)8-s + (0.121 + 0.310i)9-s + (−0.0262 − 0.350i)10-s + (−0.270 + 0.688i)11-s + (0.275 + 0.0850i)12-s + (0.486 + 0.609i)13-s + (0.509 − 0.490i)14-s + (−0.178 + 0.224i)15-s + (0.206 − 0.140i)16-s + (−1.26 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.554633 + 0.706124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554633 + 0.706124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 - 0.149i)T \) |
| 3 | \( 1 + (-0.826 - 0.563i)T \) |
| 7 | \( 1 + (2.15 - 1.52i)T \) |
| good | 5 | \( 1 + (0.0829 - 1.10i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (0.896 - 2.28i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.75 - 2.19i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (5.23 - 4.85i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.75 + 6.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.89 + 1.75i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.31 - 5.77i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.79 - 8.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.55 + 2.33i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (1.70 - 0.820i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (5.22 + 2.51i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-9.22 + 1.39i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-9.41 + 2.90i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.588 + 7.85i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-6.94 - 2.14i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.0774 + 0.134i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.31 + 10.1i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.08 - 0.315i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.92 + 3.32i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.54 + 4.45i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (4.86 + 12.4i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 8.21T + 97T^{2} \) |
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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
−11.92575736891109655796338858890, −10.75715489379484092364809754405, −10.18598881760668440240870636239, −8.925496583679257511756933282415, −8.714923056418478841026759828111, −7.04274609729775978413269930926, −6.58118476789461923383893426591, −4.97195768312279846654466101187, −3.39168922271922973956810339250, −2.20611453976296741357797466674,
0.793280827600859476037934646643, 2.71366043187353186360713614855, 3.92476532508703184217635720267, 5.71539816464596610422663083779, 6.82048119659951513529766175643, 7.84925773366486614182019817978, 8.608143649759138909965482622516, 9.601870805808658009215129942244, 10.32767800237722487504618050407, 11.49168111910147308789344596152
