L(s) = 1 | + (−22.6 + 39.1i)2-s + (−409. + 236. i)3-s + (−1.02e3 − 1.77e3i)4-s + (1.82e4 + 1.05e4i)5-s − 2.14e4i·6-s + (6.18e4 + 1.00e5i)7-s + 9.26e4·8-s + (−1.53e5 + 2.66e5i)9-s + (−8.24e5 + 4.76e5i)10-s + (−2.46e5 − 4.26e5i)11-s + (8.39e5 + 4.84e5i)12-s − 3.20e6i·13-s + (−5.32e6 + 1.59e5i)14-s − 9.95e6·15-s + (−2.09e6 + 3.63e6i)16-s + (−2.98e7 + 1.72e7i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.561 + 0.324i)3-s + (−0.249 − 0.433i)4-s + (1.16 + 0.673i)5-s − 0.458i·6-s + (0.525 + 0.850i)7-s + 0.353·8-s + (−0.289 + 0.501i)9-s + (−0.824 + 0.476i)10-s + (−0.139 − 0.241i)11-s + (0.280 + 0.162i)12-s − 0.664i·13-s + (−0.706 + 0.0211i)14-s − 0.873·15-s + (−0.125 + 0.216i)16-s + (−1.23 + 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0967i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.995 - 0.0967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0506290 + 1.04449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0506290 + 1.04449i\) |
\(L(7)\) |
|
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 | \( 1 + (-6.18e4 - 1.00e5i)T \) |
good | 3 | \( 1 + (409. - 236. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-1.82e4 - 1.05e4i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (2.46e5 + 4.26e5i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + 3.20e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (2.98e7 - 1.72e7i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (1.84e7 + 1.06e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (1.22e8 - 2.12e8i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 - 8.71e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (6.16e8 - 3.55e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-1.22e9 + 2.12e9i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 4.29e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 8.71e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (4.94e9 + 2.85e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (4.77e9 + 8.27e9i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (-4.83e10 + 2.79e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-5.97e10 - 3.45e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-3.87e10 - 6.70e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 + 1.07e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-2.07e10 + 1.19e10i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-1.59e10 + 2.76e10i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 3.39e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (-7.24e11 - 4.18e11i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 - 9.93e11iT - 6.93e23T^{2} \) |
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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
−17.63141355665508704754758534023, −15.97109330627261157697652104154, −14.73476614219214980798343361834, −13.41535128707408234136890948727, −11.20382308605302345862733988826, −10.05025821676421541129418751655, −8.388847006008858239744615071910, −6.26450552740169547007947033696, −5.25799217370831938433320716950, −2.14905473816893497243667798972,
0.52305837950325399369436557306, 1.94296593655606891994358181238, 4.64479459626078561059966717626, 6.56847151531585022432319332078, 8.741713230174260934574569009838, 10.16037500454892821880553834588, 11.61092534918068529102865037677, 12.98257918363223760036827310333, 14.14956408739827453568603512367, 16.59475076900945527142062841853