L(s) = 1 | − 0.436·5-s + i·7-s + 2.73i·11-s + 1.64i·13-s + 5.22i·17-s + 5.99·19-s − 7.68·23-s − 4.80·25-s + 4.94·29-s + 9.77i·31-s − 0.436i·35-s − 3.81i·37-s − 9.74i·41-s + 8.84·43-s + 4.54·47-s + ⋯ |
L(s) = 1 | − 0.195·5-s + 0.377i·7-s + 0.825i·11-s + 0.456i·13-s + 1.26i·17-s + 1.37·19-s − 1.60·23-s − 0.961·25-s + 0.918·29-s + 1.75i·31-s − 0.0737i·35-s − 0.627i·37-s − 1.52i·41-s + 1.34·43-s + 0.663·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9275607365\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9275607365\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 0.436T + 5T^{2} \) |
| 11 | \( 1 - 2.73iT - 11T^{2} \) |
| 13 | \( 1 - 1.64iT - 13T^{2} \) |
| 17 | \( 1 - 5.22iT - 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 9.77iT - 31T^{2} \) |
| 37 | \( 1 + 3.81iT - 37T^{2} \) |
| 41 | \( 1 + 9.74iT - 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 - 13.2iT - 59T^{2} \) |
| 61 | \( 1 + 6.26iT - 61T^{2} \) |
| 67 | \( 1 + 9.42T + 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 0.227iT - 83T^{2} \) |
| 89 | \( 1 + 2.46iT - 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−8.436754782235299455703730731906, −7.53104225480611220982910423675, −7.19894726333928426802509231219, −6.06151417599661252307220368759, −5.74067864716546393042804108014, −4.65887544215289003964525961030, −4.06705455962787712031698670251, −3.22627689412645147787981627331, −2.15984693792710391208844639719, −1.42467786521355378012279491194,
0.24583708405724929511078954041, 1.20410448213283119308899493518, 2.55243988768263743836440909189, 3.22267088962048433393426324208, 4.09276927842889652427664045543, 4.81040761556321421756262400998, 5.78832311873814182602782603807, 6.15218695812048383259208625991, 7.22793859862324064809027901531, 7.85941383150831786991551124578