| L(s) = 1 | − 3.41·3-s − 2.23i·5-s − 11.7i·7-s + 2.67·9-s − 1.72i·11-s + 16.3·13-s + 7.64i·15-s − 6.55i·17-s − 31.1i·19-s + 40.1i·21-s + (−16.3 − 16.1i)23-s − 5.00·25-s + 21.6·27-s + 53.4·29-s + 40.6·31-s + ⋯ |
| L(s) = 1 | − 1.13·3-s − 0.447i·5-s − 1.67i·7-s + 0.297·9-s − 0.156i·11-s + 1.26·13-s + 0.509i·15-s − 0.385i·17-s − 1.63i·19-s + 1.91i·21-s + (−0.712 − 0.701i)23-s − 0.200·25-s + 0.800·27-s + 1.84·29-s + 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.396434276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396434276\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (16.3 + 16.1i)T \) |
| good | 3 | \( 1 + 3.41T + 9T^{2} \) |
| 7 | \( 1 + 11.7iT - 49T^{2} \) |
| 11 | \( 1 + 1.72iT - 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 + 6.55iT - 289T^{2} \) |
| 19 | \( 1 + 31.1iT - 361T^{2} \) |
| 29 | \( 1 - 53.4T + 841T^{2} \) |
| 31 | \( 1 - 40.6T + 961T^{2} \) |
| 37 | \( 1 + 48.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 71.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 13.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 37.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 76.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 16.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 76.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 77.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 71.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 96.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 111. iT - 9.40e3T^{2} \) |
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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
−8.705903057448988512696766609584, −7.982543304703296475935077037800, −6.90954023216484484688953701398, −6.47246286280305799353447111723, −5.57155175459628109473912015691, −4.52791322902993180926334646550, −4.16457580771166114990872577338, −2.77157178874845020231295376918, −0.844388930552770690660950747015, −0.68882126030848559700868619098,
1.19075304443739906782436706505, 2.41753036755735229674108927446, 3.45874246459826624631937381118, 4.63967443044788975310161040008, 5.64043460702595299465737988734, 6.12241858345593397951694932322, 6.47349674380594227456513918586, 8.040477295272459903408677933691, 8.397044196724959146009441971696, 9.437993565916512162427199122807
