| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.23i·5-s + 3.80i·7-s − 2.82·8-s − 3.16i·10-s + 16.1i·11-s + 12.9·13-s − 5.38i·14-s + 4.00·16-s + 10.8i·17-s + 17.1i·19-s + 4.47i·20-s − 22.8i·22-s + (−11.6 + 19.8i)23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.447i·5-s + 0.543i·7-s − 0.353·8-s − 0.316i·10-s + 1.47i·11-s + 0.997·13-s − 0.384i·14-s + 0.250·16-s + 0.636i·17-s + 0.900i·19-s + 0.223i·20-s − 1.04i·22-s + (−0.506 + 0.862i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.198434076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198434076\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (11.6 - 19.8i)T \) |
| good | 7 | \( 1 - 3.80iT - 49T^{2} \) |
| 11 | \( 1 - 16.1iT - 121T^{2} \) |
| 13 | \( 1 - 12.9T + 169T^{2} \) |
| 17 | \( 1 - 10.8iT - 289T^{2} \) |
| 19 | \( 1 - 17.1iT - 361T^{2} \) |
| 29 | \( 1 - 10.1T + 841T^{2} \) |
| 31 | \( 1 - 29.2T + 961T^{2} \) |
| 37 | \( 1 - 3.06iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 55.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 28.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 52.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 7.48iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 121.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 132.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 18.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 46.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.4iT - 9.40e3T^{2} \) |
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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
−9.335056985552122041782135385578, −8.392398657809276492352821043331, −7.88429088640699354444531247062, −6.95782998613043140604858854946, −6.27266944035324525454761999013, −5.49507304513782596128878665068, −4.28572459076656603949930498885, −3.35169495617340967090765152536, −2.18854828259761939083541853808, −1.41533190638318530391138300093,
0.43807602688594242569036001491, 1.10208347891603303287580632229, 2.56540940675843574202070549924, 3.52278335995477132486225357147, 4.50335806371584625835962448767, 5.60203228373578805348584665795, 6.34434073039894401439074376844, 7.11439053741184734092039709919, 8.082998122588613314871867367904, 8.683184542779588198007778200930
