L(s) = 1 | + (6.35 − 6.35i)2-s + (0.342 + 140. i)3-s + 431. i·4-s + (−1.05e3 − 919. i)5-s + (893. + 889. i)6-s + (−7.86e3 − 7.86e3i)7-s + (5.99e3 + 5.99e3i)8-s + (−1.96e4 + 96.0i)9-s + (−1.25e4 + 839. i)10-s + 5.07e4i·11-s + (−6.04e4 + 147. i)12-s + (−4.94e3 + 4.94e3i)13-s − 9.99e4·14-s + (1.28e5 − 1.47e5i)15-s − 1.44e5·16-s + (−1.74e5 + 1.74e5i)17-s + ⋯ |
L(s) = 1 | + (0.280 − 0.280i)2-s + (0.00243 + 0.999i)3-s + 0.842i·4-s + (−0.752 − 0.658i)5-s + (0.281 + 0.280i)6-s + (−1.23 − 1.23i)7-s + (0.517 + 0.517i)8-s + (−0.999 + 0.00487i)9-s + (−0.396 + 0.0265i)10-s + 1.04i·11-s + (−0.842 + 0.00205i)12-s + (−0.0480 + 0.0480i)13-s − 0.695·14-s + (0.656 − 0.754i)15-s − 0.551·16-s + (−0.507 + 0.507i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.0436933 + 0.536688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0436933 + 0.536688i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.342 - 140. i)T \) |
| 5 | \( 1 + (1.05e3 + 919. i)T \) |
good | 2 | \( 1 + (-6.35 + 6.35i)T - 512iT^{2} \) |
| 7 | \( 1 + (7.86e3 + 7.86e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 - 5.07e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (4.94e3 - 4.94e3i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (1.74e5 - 1.74e5i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 3.60e4iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (-9.95e5 - 9.95e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 - 1.37e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.36e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (4.18e6 + 4.18e6i)T + 1.29e14iT^{2} \) |
| 41 | \( 1 - 1.69e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (-4.04e6 + 4.04e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-3.93e7 + 3.93e7i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (-4.34e7 - 4.34e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + 1.26e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.56e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + (7.00e7 + 7.00e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 + 3.78e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.07e8 + 1.07e8i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 - 6.60e7iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (3.05e8 + 3.05e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 - 4.49e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.77e8 + 1.77e8i)T + 7.60e17iT^{2} \) |
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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
−17.19095454427168824350333113511, −16.52861213375434323453868277389, −15.39648911489713422707395786488, −13.43845668516149078223897932554, −12.29986408321535782742557226706, −10.73304257791744810850289928291, −9.173714265280191819795515625105, −7.37602071706597657884295976331, −4.46389247917359218091523782744, −3.50196931198950899694999990779,
0.24644143997199041334399091954, 2.81992434669930754527244503207, 5.85316934331565264126620218834, 6.89318754462180670469673449134, 8.927888807525103548494476663136, 11.01000508897486414710089415627, 12.44798190825503902997425963380, 13.82837766043971780716249445031, 15.14541879231397758206991882389, 16.18558451892084511429768125968