| 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
−8.635206160167547645352611683963, −7.75578452465452939134392011862, −6.95486517140137659624673206522, −6.39265695515949371628127647057, −5.35544312205131889319749782822, −4.40183455781818229239922049676, −4.07809651890719114961633882846, −2.79310678777461091084819409455, −1.40055863120825409765808011288, −0.69740321406803353008084554793,
1.57489083333835240207871226755, 2.39819863389881099425942861282, 3.30618394724812572464092514897, 4.19063964880481099421733463882, 5.46398715315674357324730757266, 5.98609390835085804044246900868, 6.56942591861431131559323606230, 7.44983184432701905815571031317, 8.387915034836467193652106471304, 9.129214036912626185622187726035
