| L(s) = 1 | + (1 + 2i)5-s + 4i·7-s + 4·11-s + 4i·17-s + 4i·23-s + (−3 + 4i)25-s + 6·29-s − 4·31-s + (−8 + 4i)35-s − 8i·37-s + 10·41-s + 4i·43-s + 4i·47-s − 9·49-s − 12i·53-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.894i)5-s + 1.51i·7-s + 1.20·11-s + 0.970i·17-s + 0.834i·23-s + (−0.600 + 0.800i)25-s + 1.11·29-s − 0.718·31-s + (−1.35 + 0.676i)35-s − 1.31i·37-s + 1.56·41-s + 0.609i·43-s + 0.583i·47-s − 1.28·49-s − 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.048763175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.048763175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.129214036912626185622187726035, −8.387915034836467193652106471304, −7.44983184432701905815571031317, −6.56942591861431131559323606230, −5.98609390835085804044246900868, −5.46398715315674357324730757266, −4.19063964880481099421733463882, −3.30618394724812572464092514897, −2.39819863389881099425942861282, −1.57489083333835240207871226755,
0.69740321406803353008084554793, 1.40055863120825409765808011288, 2.79310678777461091084819409455, 4.07809651890719114961633882846, 4.40183455781818229239922049676, 5.35544312205131889319749782822, 6.39265695515949371628127647057, 6.95486517140137659624673206522, 7.75578452465452939134392011862, 8.635206160167547645352611683963
