| L(s) = 1 | + (−40.5 + 70.1i)3-s + (−519. − 899. i)5-s + (−5.77e3 + 2.64e3i)7-s + (−3.28e3 − 5.68e3i)9-s + (−2.47e4 + 4.28e4i)11-s + 1.34e5·13-s + 8.40e4·15-s + (2.53e5 − 4.38e5i)17-s + (4.75e5 + 8.23e5i)19-s + (4.79e4 − 5.12e5i)21-s + (−4.36e5 − 7.55e5i)23-s + (4.37e5 − 7.58e5i)25-s + 5.31e5·27-s − 1.12e6·29-s + (−8.85e5 + 1.53e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.500i)3-s + (−0.371 − 0.643i)5-s + (−0.908 + 0.417i)7-s + (−0.166 − 0.288i)9-s + (−0.509 + 0.881i)11-s + 1.30·13-s + 0.428·15-s + (0.735 − 1.27i)17-s + (0.836 + 1.44i)19-s + (0.0538 − 0.574i)21-s + (−0.324 − 0.562i)23-s + (0.224 − 0.388i)25-s + 0.192·27-s − 0.295·29-s + (−0.172 + 0.298i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4216367606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4216367606\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (40.5 - 70.1i)T \) |
| 7 | \( 1 + (5.77e3 - 2.64e3i)T \) |
| good | 5 | \( 1 + (519. + 899. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (2.47e4 - 4.28e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.34e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-2.53e5 + 4.38e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.75e5 - 8.23e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (4.36e5 + 7.55e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (8.85e5 - 1.53e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-2.51e6 - 4.35e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 2.26e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.75e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (8.13e6 + 1.40e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (1.96e7 - 3.39e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (7.10e7 - 1.23e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-8.11e7 - 1.40e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.79e7 + 1.52e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 3.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-7.28e7 + 1.26e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (7.60e7 + 1.31e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 7.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-3.06e7 - 5.30e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.09e9T + 7.60e17T^{2} \) |
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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
−11.78766540079784205017861179088, −10.38642117667307265994787687224, −9.657329304693211799474383633080, −8.667073899676713251796117547815, −7.53302808390434093792624047470, −6.15013129796577385210711610229, −5.21830613101496555797508178974, −4.03063730780402489229226809125, −2.92846256185562363584056524612, −1.16244653987652446602746957468,
0.11734970312909151353673541092, 1.23052069254993288544739681967, 3.00033970800767819729553528319, 3.72890295528763255768209061785, 5.59426957968971775099537417215, 6.43294460648483490143784676781, 7.41405533864587502736488029646, 8.414023144629132146903225209269, 9.723402573459513577038327498725, 10.95864718270253357919422836427
