| L(s) = 1 | + (−16 − 27.7i)2-s + (−511. + 886. i)4-s + (−1.60e3 − 2.77e3i)5-s + (3.30e4 − 2.96e4i)7-s + 3.27e4·8-s + (−5.12e4 + 8.87e4i)10-s + (5.14e5 − 8.90e5i)11-s − 1.32e6·13-s + (−1.35e6 − 4.42e5i)14-s + (−5.24e5 − 9.08e5i)16-s + (6.26e5 − 1.08e6i)17-s + (−2.39e6 − 4.15e6i)19-s + 3.28e6·20-s − 3.28e7·22-s + (4.72e6 + 8.18e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.229 − 0.397i)5-s + (0.744 − 0.667i)7-s + 0.353·8-s + (−0.162 + 0.280i)10-s + (0.962 − 1.66i)11-s − 0.990·13-s + (−0.672 − 0.219i)14-s + (−0.125 − 0.216i)16-s + (0.106 − 0.185i)17-s + (−0.222 − 0.384i)19-s + 0.229·20-s − 1.36·22-s + (0.153 + 0.265i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.677683622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677683622\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-3.30e4 + 2.96e4i)T \) |
| good | 5 | \( 1 + (1.60e3 + 2.77e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-5.14e5 + 8.90e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.32e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-6.26e5 + 1.08e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (2.39e6 + 4.15e6i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-4.72e6 - 8.18e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.54e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-7.14e7 + 1.23e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-6.49e7 - 1.12e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 2.27e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.66e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-8.69e8 - 1.50e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (6.46e8 - 1.11e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-4.51e9 + 7.81e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (3.00e9 + 5.19e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.96e9 + 3.41e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.31e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (2.31e9 - 4.00e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-2.40e10 - 4.15e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 2.42e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.81e10 + 8.33e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.29e11T + 7.15e21T^{2} \) |
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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
−10.93890813442032148361816836621, −9.747026958358996213881134504243, −8.642198263524098606552466935297, −7.899455832092818010922901917509, −6.53585291179619164692368707574, −4.93521730831081783994929756872, −3.95483430408905952125746674443, −2.67304991053359531696321580068, −1.12028841487385658333271173606, −0.49381450396452801618437969878,
1.31478490369659245847932833912, 2.44434815032180263947847977348, 4.28738224340966732317438199548, 5.19461320495874066152784834506, 6.65359585559810408401750055812, 7.40016612032090772880871297594, 8.536847317539278735607022814604, 9.556308479883149129016544324674, 10.50187446970201811979536119672, 11.87338577923161133869080893903
