| L(s) = 1 | + 0.530·5-s − i·7-s − 0.905i·11-s + 4.38i·13-s − 1.25i·17-s − 0.00364·19-s + 0.778·23-s − 4.71·25-s − 5.43·29-s − 3.49i·31-s − 0.530i·35-s + 5.34i·37-s + 1.74i·41-s + 0.259·43-s − 3.75·47-s + ⋯ |
| L(s) = 1 | + 0.237·5-s − 0.377i·7-s − 0.272i·11-s + 1.21i·13-s − 0.304i·17-s − 0.000837·19-s + 0.162·23-s − 0.943·25-s − 1.00·29-s − 0.628i·31-s − 0.0895i·35-s + 0.878i·37-s + 0.272i·41-s + 0.0395·43-s − 0.548·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3538039088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3538039088\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.530T + 5T^{2} \) |
| 11 | \( 1 + 0.905iT - 11T^{2} \) |
| 13 | \( 1 - 4.38iT - 13T^{2} \) |
| 17 | \( 1 + 1.25iT - 17T^{2} \) |
| 19 | \( 1 + 0.00364T + 19T^{2} \) |
| 23 | \( 1 - 0.778T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 + 3.49iT - 31T^{2} \) |
| 37 | \( 1 - 5.34iT - 37T^{2} \) |
| 41 | \( 1 - 1.74iT - 41T^{2} \) |
| 43 | \( 1 - 0.259T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 - 5.73iT - 59T^{2} \) |
| 61 | \( 1 + 0.555iT - 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.96iT - 79T^{2} \) |
| 83 | \( 1 + 5.82iT - 83T^{2} \) |
| 89 | \( 1 + 1.94iT - 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.355999756227845596659904480972, −7.62115099153638979743297314075, −6.98088009948984916595786821765, −6.26798851030935431337627335058, −5.61484347442625739311648875827, −4.67252122728388433751228606555, −4.06128349372184000382647150198, −3.20607994459019992695459755057, −2.17537360087159640821452014766, −1.34257466730415759926589683102,
0.085557680578157647393934796761, 1.47778456949950545919898782629, 2.37252667299442775171385584220, 3.26497149860847258047082019622, 4.02271568344902427185908039098, 5.05738751567382590683694426336, 5.60804298643063283543429684123, 6.22792074912460878862169017949, 7.13753654988676134156735418917, 7.80358595594432312717388401560
