| L(s) = 1 | + (16 + 27.7i)2-s + (−511. + 886. i)4-s + (−2.02e3 − 3.50e3i)5-s + (−4.12e4 − 1.66e4i)7-s − 3.27e4·8-s + (6.48e4 − 1.12e5i)10-s + (−5.26e4 + 9.11e4i)11-s + 1.44e5·13-s + (−1.98e5 − 1.40e6i)14-s + (−5.24e5 − 9.08e5i)16-s + (3.54e6 − 6.14e6i)17-s + (2.86e6 + 4.96e6i)19-s + 4.14e6·20-s − 3.36e6·22-s + (−1.91e7 − 3.31e7i)23-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.289 − 0.502i)5-s + (−0.927 − 0.374i)7-s − 0.353·8-s + (0.205 − 0.355i)10-s + (−0.0985 + 0.170i)11-s + 0.108·13-s + (−0.0985 − 0.700i)14-s + (−0.125 − 0.216i)16-s + (0.605 − 1.04i)17-s + (0.265 + 0.460i)19-s + 0.289·20-s − 0.139·22-s + (−0.619 − 1.07i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.016701705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.016701705\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 - 27.7i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4.12e4 + 1.66e4i)T \) |
| good | 5 | \( 1 + (2.02e3 + 3.50e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (5.26e4 - 9.11e4i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.44e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-3.54e6 + 6.14e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-2.86e6 - 4.96e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.91e7 + 3.31e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.21e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (5.57e7 - 9.65e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.38e8 - 2.39e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.41e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.35e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-3.25e7 - 5.63e7i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (2.08e9 - 3.60e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-4.30e8 + 7.45e8i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.59e8 + 4.49e8i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (7.81e9 - 1.35e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 6.55e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (4.05e9 - 7.02e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.72e10 - 2.98e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 1.62e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.95e10 - 3.38e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.29e10T + 7.15e21T^{2} \) |
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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
−11.99426949838306992885076102375, −10.42960902593103825430756066003, −9.425115272759943030585269350094, −8.305716382641716063113648497558, −7.25281945055601137552641503389, −6.26221160811893754447989840616, −5.04560663148275439325087674369, −3.96453694381847746434424638577, −2.80924220213366517693794338947, −0.881325910465644355221311248214,
0.24446416174055826914943448962, 1.77206345431418180397143153171, 3.10362598617126381307187106159, 3.78733381866688219883538309580, 5.41439726762311684534583772671, 6.37380143555937262539534173257, 7.66435942559567445068052644812, 9.059827450658158368234302707211, 9.995638629433101525406692864625, 10.96761117026176391172663895651
