| L(s) = 1 | + (5.65 − 9.79i)2-s + (−63.9 − 110. i)4-s + (−753. − 435. i)5-s + (−1.83e3 + 1.54e3i)7-s − 1.44e3·8-s + (−8.52e3 + 4.92e3i)10-s + (−1.43e4 − 2.48e4i)11-s − 3.03e4i·13-s + (4.77e3 + 2.67e4i)14-s + (−8.19e3 + 1.41e4i)16-s + (−3.69e4 + 2.13e4i)17-s + (7.34e3 + 4.24e3i)19-s + 1.11e5i·20-s − 3.24e5·22-s + (−1.47e5 + 2.56e5i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.20 − 0.696i)5-s + (−0.764 + 0.644i)7-s − 0.353·8-s + (−0.852 + 0.492i)10-s + (−0.980 − 1.69i)11-s − 1.06i·13-s + (0.124 + 0.696i)14-s + (−0.125 + 0.216i)16-s + (−0.442 + 0.255i)17-s + (0.0563 + 0.0325i)19-s + 0.696i·20-s − 1.38·22-s + (−0.528 + 0.914i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2596784004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2596784004\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 + 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.83e3 - 1.54e3i)T \) |
| good | 5 | \( 1 + (753. + 435. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.43e4 + 2.48e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 3.03e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (3.69e4 - 2.13e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-7.34e3 - 4.24e3i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.47e5 - 2.56e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 7.52e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.71e5 + 9.89e4i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-5.82e5 + 1.00e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 5.27e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.25e4T + 1.16e13T^{2} \) |
| 47 | \( 1 + (5.49e6 + 3.17e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.10e5 + 5.37e5i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-6.37e6 + 3.68e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.38e7 - 8.01e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.07e6 - 1.86e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 6.33e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-6.27e5 + 3.62e5i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-5.35e6 + 9.28e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 6.28e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.63e7 - 9.43e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 2.44e7iT - 7.83e15T^{2} \) |
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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
−11.90426087112808154947825294674, −11.13697016130238659941319088135, −9.982582530458509535332508151362, −8.589128374750182184558437466677, −8.007246051039280426875698197649, −6.08832886705019617029766449976, −5.10564453078181758344893877993, −3.64145154583511855258324807299, −2.82551866219814506139769722517, −0.74744518181260869884404765744,
0.089587571819913779118041373307, 2.53413054789907055775704797234, 3.92992189661203042349852265323, 4.70944235846089466047614890517, 6.68605615440189081045890844957, 7.12880965573581822884597874069, 8.142936048171085729744453752122, 9.680772673314287346648275905588, 10.67788042513417984659974261223, 11.94329427665839334054693730436
