L(s) = 1 | + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (4.29 − 2.55i)5-s + 2.44·6-s + (4.10 + 4.10i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−6.85 − 1.74i)10-s − 12.9·11-s + (−2.44 − 2.44i)12-s + (−7.48 + 7.48i)13-s − 8.20i·14-s + (−2.13 + 8.39i)15-s − 4·16-s + (−23.2 − 23.2i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.859 − 0.510i)5-s + 0.408·6-s + (0.585 + 0.585i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.685 − 0.174i)10-s − 1.17·11-s + (−0.204 − 0.204i)12-s + (−0.576 + 0.576i)13-s − 0.585i·14-s + (−0.142 + 0.559i)15-s − 0.250·16-s + (−1.36 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3614957559\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3614957559\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (-4.29 + 2.55i)T \) |
| 23 | \( 1 + (3.39 - 3.39i)T \) |
good | 7 | \( 1 + (-4.10 - 4.10i)T + 49iT^{2} \) |
| 11 | \( 1 + 12.9T + 121T^{2} \) |
| 13 | \( 1 + (7.48 - 7.48i)T - 169iT^{2} \) |
| 17 | \( 1 + (23.2 + 23.2i)T + 289iT^{2} \) |
| 19 | \( 1 - 2.66iT - 361T^{2} \) |
| 29 | \( 1 + 13.0iT - 841T^{2} \) |
| 31 | \( 1 - 21.6T + 961T^{2} \) |
| 37 | \( 1 + (1.73 + 1.73i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 41.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (2.10 - 2.10i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (2.10 + 2.10i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (8.58 - 8.58i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 65.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 39.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + (42.4 + 42.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 119.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (86.2 - 86.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 63.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-98.1 + 98.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 156. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-49.2 - 49.2i)T + 9.40e3iT^{2} \) |
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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
−9.890303284640419436543849745394, −9.148272651594609782679924579330, −8.481040257616319297447063548699, −7.35876850830471669053364335007, −6.20382629234574709310267301816, −5.04916802517038796650003079647, −4.63179174900909655691355018056, −2.75354649213804510099118372384, −1.90415697811472354436888136128, −0.15057510042246014083898488437,
1.55168285521832261811422977367, 2.65844496480435452647620772612, 4.55323921802495064045901284902, 5.47674009387560286663955808537, 6.32714269256786217309914118993, 7.14983792380132542055993206635, 7.939108568116480979826080091612, 8.762894070853911957911479302795, 10.03654159166187760066778013982, 10.58387936305119840342963931577