| L(s) = 1 | + (22.6 + 39.1i)2-s + (444. + 256. i)3-s + (−1.02e3 + 1.77e3i)4-s + (1.26e4 − 7.32e3i)5-s + 2.32e4i·6-s − 9.26e4·8-s + (−1.34e5 − 2.32e5i)9-s + (5.74e5 + 3.31e5i)10-s + (1.34e6 − 2.32e6i)11-s + (−9.09e5 + 5.25e5i)12-s + 1.34e6i·13-s + 7.51e6·15-s + (−2.09e6 − 3.63e6i)16-s + (−6.20e6 − 3.58e6i)17-s + (6.07e6 − 1.05e7i)18-s + (−6.33e7 + 3.65e7i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.609 + 0.351i)3-s + (−0.249 + 0.433i)4-s + (0.812 − 0.468i)5-s + 0.497i·6-s − 0.353·8-s + (−0.252 − 0.437i)9-s + (0.574 + 0.331i)10-s + (0.757 − 1.31i)11-s + (−0.304 + 0.175i)12-s + 0.279i·13-s + 0.659·15-s + (−0.125 − 0.216i)16-s + (−0.257 − 0.148i)17-s + (0.178 − 0.309i)18-s + (−1.34 + 0.777i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.535605644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535605644\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-22.6 - 39.1i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-444. - 256. i)T + (2.65e5 + 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-1.26e4 + 7.32e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-1.34e6 + 2.32e6i)T + (-1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 - 1.34e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (6.20e6 + 3.58e6i)T + (2.91e14 + 5.04e14i)T^{2} \) |
| 19 | \( 1 + (6.33e7 - 3.65e7i)T + (1.10e15 - 1.91e15i)T^{2} \) |
| 23 | \( 1 + (1.14e8 + 1.97e8i)T + (-1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 + 5.69e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (2.67e8 + 1.54e8i)T + (3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + (8.51e8 + 1.47e9i)T + (-3.29e18 + 5.70e18i)T^{2} \) |
| 41 | \( 1 - 5.37e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 9.19e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (9.45e9 - 5.45e9i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-7.92e9 + 1.37e10i)T + (-2.45e20 - 4.25e20i)T^{2} \) |
| 59 | \( 1 + (1.75e10 + 1.01e10i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (2.38e10 - 1.37e10i)T + (1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (3.35e10 - 5.80e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 - 7.34e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (5.62e10 + 3.24e10i)T + (1.14e22 + 1.98e22i)T^{2} \) |
| 79 | \( 1 + (3.16e9 + 5.48e9i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 - 5.07e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (-5.49e11 + 3.17e11i)T + (1.23e23 - 2.13e23i)T^{2} \) |
| 97 | \( 1 + 3.89e11iT - 6.93e23T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.33749785560258915244831838332, −9.852190173848648700319201503253, −8.865652279742821953085755728848, −8.306942808602509852786222608329, −6.46417664810937061208467978794, −5.83955578756559731164124837386, −4.33005134722967371430949213711, −3.34738007149404112004845489028, −1.87572326344976325164227006377, −0.23517717081877376459944666186,
1.82093854553656086852417222423, 2.11729514424979753293594800593, 3.53380063068215068318969999268, 4.88959399130029345885736581426, 6.23074740395825742392038820941, 7.36323724107481010925236722627, 8.790678064909014620677624711842, 9.782724125229889352744505292591, 10.70470958333241844603038269246, 11.88660134652094467293994838407
