| L(s) = 1 | + (−0.699 + 1.21i)2-s + (5.27 − 9.14i)3-s + (15.0 + 26.0i)4-s + (26.3 − 45.6i)5-s + (7.38 + 12.7i)6-s + 195.·7-s − 86.8·8-s + (65.7 + 113. i)9-s + (36.8 + 63.8i)10-s − 538.·11-s + 317.·12-s + (−456. − 790. i)13-s + (−137. + 237. i)14-s + (−278. − 482. i)15-s + (−419. + 727. i)16-s + (−381. + 660. i)17-s + ⋯ |
| L(s) = 1 | + (−0.123 + 0.214i)2-s + (0.338 − 0.586i)3-s + (0.469 + 0.813i)4-s + (0.471 − 0.816i)5-s + (0.0837 + 0.145i)6-s + 1.51·7-s − 0.479·8-s + (0.270 + 0.468i)9-s + (0.116 + 0.201i)10-s − 1.34·11-s + 0.635·12-s + (−0.749 − 1.29i)13-s + (−0.186 + 0.323i)14-s + (−0.319 − 0.553i)15-s + (−0.410 + 0.710i)16-s + (−0.319 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.69188 - 0.0151421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69188 - 0.0151421i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-454. - 1.50e3i)T \) |
| good | 2 | \( 1 + (0.699 - 1.21i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (-5.27 + 9.14i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-26.3 + 45.6i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 538.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (456. + 790. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (381. - 660. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (1.34e3 + 2.32e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (845. + 1.46e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 5.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 499.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (5.55e3 - 9.62e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-2.44e3 + 4.23e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (340. + 589. i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (339. + 587. i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.44e4 + 2.50e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-7.70e3 - 1.33e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.04e4 - 1.81e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.52e4 + 6.10e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.66e4 - 2.89e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.16e4 + 5.47e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 7.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-7.29e4 - 1.26e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.44e4 - 2.49e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−17.52674396467833269806955481744, −16.35626253459161325978200873939, −14.88866491291026343550806668302, −13.22294283715725718962215275789, −12.41572558544044733550900747268, −10.63056203679200393137802465210, −8.211530379449297208322835159684, −7.76943943042857791170258834480, −5.18011197731708727817375079676, −2.10195578167585347449036396417,
2.23702350605051611373381880345, 5.04956976070911395706100992532, 7.12943847811133901810698112686, 9.340301553542869661402472813201, 10.55113986527193890872283156548, 11.56390015374837156637675956834, 14.03157317299275479435499543154, 14.80624743984342578228706292593, 15.80052408177356347013574268096, 17.86006838354200356491762596239
