L(s) = 1 | + (−1.87 − 0.696i)2-s − 1.73i·3-s + (3.02 + 2.61i)4-s + (−1.20 + 3.24i)6-s + 5.46i·7-s + (−3.86 − 7.00i)8-s − 2.99·9-s − 11.0i·11-s + (4.52 − 5.24i)12-s − 10.1·13-s + (3.80 − 10.2i)14-s + (2.35 + 15.8i)16-s + 24.4·17-s + (5.62 + 2.08i)18-s − 23.7i·19-s + ⋯ |
L(s) = 1 | + (−0.937 − 0.348i)2-s − 0.577i·3-s + (0.757 + 0.652i)4-s + (−0.201 + 0.541i)6-s + 0.781i·7-s + (−0.482 − 0.875i)8-s − 0.333·9-s − 1.00i·11-s + (0.376 − 0.437i)12-s − 0.778·13-s + (0.272 − 0.732i)14-s + (0.147 + 0.989i)16-s + 1.43·17-s + (0.312 + 0.116i)18-s − 1.25i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.311851 - 0.680598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.311851 - 0.680598i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.87 + 0.696i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5.46iT - 49T^{2} \) |
| 11 | \( 1 + 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 10.1T + 169T^{2} \) |
| 17 | \( 1 - 24.4T + 289T^{2} \) |
| 19 | \( 1 + 23.7iT - 361T^{2} \) |
| 23 | \( 1 + 37.2iT - 529T^{2} \) |
| 29 | \( 1 + 25.7T + 841T^{2} \) |
| 31 | \( 1 - 4.83iT - 961T^{2} \) |
| 37 | \( 1 + 35.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 9.30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 70.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 55.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 104. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 76.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 93.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 49.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 72.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 115.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 72.9T + 9.40e3T^{2} \) |
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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
−11.17087121635146803873204440199, −10.25608517084672215896607425250, −9.108249722022896309444834061108, −8.474277739671944668002032592793, −7.47879797880192651766998477262, −6.49905229362027337743507333896, −5.33679400231403273060460407094, −3.28019375946473253548931554382, −2.21806415804789340045358297050, −0.49484728645730302719364501048,
1.57939099203242462227164767656, 3.43467498262371742651199555536, 4.93646776777897967380199884982, 6.00909300018571437243018757774, 7.50024178529091049494812742895, 7.73449956921370032826419279608, 9.348758789335094981315371759407, 9.876891244180425250369548957118, 10.53297691517109449464682282396, 11.64804796159972554974986643881