| L(s) = 1 | + 1.08e5i·2-s + (−2.05e9 + 2.81e9i)3-s + 1.08e12·4-s + 1.13e14i·5-s + (−3.04e14 − 2.21e14i)6-s − 1.19e17·7-s + 2.36e17i·8-s + (−3.72e18 − 1.15e19i)9-s − 1.22e19·10-s + 1.08e21i·11-s + (−2.23e21 + 3.06e21i)12-s + 9.34e21·13-s − 1.28e22i·14-s + (−3.19e23 − 2.33e23i)15-s + 1.17e24·16-s + 1.14e24i·17-s + ⋯ |
| L(s) = 1 | + 0.103i·2-s + (−0.588 + 0.808i)3-s + 0.989·4-s + 1.19i·5-s + (−0.0833 − 0.0606i)6-s − 1.49·7-s + 0.205i·8-s + (−0.306 − 0.951i)9-s − 0.122·10-s + 1.60i·11-s + (−0.582 + 0.799i)12-s + 0.491·13-s − 0.153i·14-s + (−0.961 − 0.700i)15-s + 0.968·16-s + 0.281i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{41}{2})\) |
\(\approx\) |
\(0.6577267601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6577267601\) |
| \(L(21)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.05e9 - 2.81e9i)T \) |
| good | 2 | \( 1 - 1.08e5iT - 1.09e12T^{2} \) |
| 5 | \( 1 - 1.13e14iT - 9.09e27T^{2} \) |
| 7 | \( 1 + 1.19e17T + 6.36e33T^{2} \) |
| 11 | \( 1 - 1.08e21iT - 4.52e41T^{2} \) |
| 13 | \( 1 - 9.34e21T + 3.61e44T^{2} \) |
| 17 | \( 1 - 1.14e24iT - 1.65e49T^{2} \) |
| 19 | \( 1 + 4.02e25T + 1.41e51T^{2} \) |
| 23 | \( 1 + 1.54e26iT - 2.94e54T^{2} \) |
| 29 | \( 1 + 3.32e29iT - 3.13e58T^{2} \) |
| 31 | \( 1 + 2.50e29T + 4.51e59T^{2} \) |
| 37 | \( 1 + 2.26e31T + 5.34e62T^{2} \) |
| 41 | \( 1 + 2.34e32iT - 3.24e64T^{2} \) |
| 43 | \( 1 - 8.11e31T + 2.18e65T^{2} \) |
| 47 | \( 1 - 2.91e33iT - 7.65e66T^{2} \) |
| 53 | \( 1 + 1.51e34iT - 9.35e68T^{2} \) |
| 59 | \( 1 + 7.34e34iT - 6.82e70T^{2} \) |
| 61 | \( 1 + 1.65e35T + 2.58e71T^{2} \) |
| 67 | \( 1 - 8.14e35T + 1.10e73T^{2} \) |
| 71 | \( 1 - 3.89e36iT - 1.12e74T^{2} \) |
| 73 | \( 1 + 1.48e37T + 3.41e74T^{2} \) |
| 79 | \( 1 - 5.20e37T + 8.03e75T^{2} \) |
| 83 | \( 1 - 2.17e38iT - 5.79e76T^{2} \) |
| 89 | \( 1 - 3.50e38iT - 9.45e77T^{2} \) |
| 97 | \( 1 + 7.87e39T + 2.95e79T^{2} \) |
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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
−17.29432161004492455970581973039, −15.77632446795323356689121038118, −14.98485071511221232632293461898, −12.37188649702279391599748061039, −10.76858557376150241042227765768, −9.858099627318245593465314562389, −6.93523549291879920695739071535, −6.15031071182915913469636883834, −3.78279941035283154121965713196, −2.42838845945002052476658950598,
0.21580314364448787219176993925, 1.33658323555982520019401538057, 3.14428567448025920879624030926, 5.68445795166356836946990698394, 6.70873098676283684420628478134, 8.618268503238157929399085227645, 10.83345492369442319173333448162, 12.33618651699729235289863077363, 13.27436991091478476989140288932, 16.23581484056005738998310388186
