L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.63 + 0.189i)7-s − 0.999·8-s + (3.44 − 1.99i)11-s + 0.0681·13-s + (1.48 − 2.19i)14-s + (−0.5 + 0.866i)16-s + (6.34 − 3.66i)17-s + (1.76 + 1.01i)19-s − 3.98i·22-s + (−1.86 + 3.23i)23-s + (0.0340 − 0.0590i)26-s + (−1.15 − 2.38i)28-s + 0.898i·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.997 + 0.0716i)7-s − 0.353·8-s + (1.03 − 0.600i)11-s + 0.0189·13-s + (0.396 − 0.585i)14-s + (−0.125 + 0.216i)16-s + (1.53 − 0.888i)17-s + (0.404 + 0.233i)19-s − 0.848i·22-s + (−0.389 + 0.673i)23-s + (0.00668 − 0.0115i)26-s + (−0.218 − 0.449i)28-s + 0.166i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.801846888\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.801846888\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.189i)T \) |
good | 11 | \( 1 + (-3.44 + 1.99i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.0681T + 13T^{2} \) |
| 17 | \( 1 + (-6.34 + 3.66i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 1.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.86 - 3.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.898iT - 29T^{2} \) |
| 31 | \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.52 + 2.03i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 + 0.964iT - 43T^{2} \) |
| 47 | \( 1 + (-1.43 - 0.830i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.61 - 11.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.32 - 9.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 - 3.76i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.23 - 5.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.93iT - 71T^{2} \) |
| 73 | \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.77 + 15.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (-0.913 + 1.58i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.803226346972639176435422430965, −7.68944890909445013139817218892, −7.22719956503191091396495194739, −5.88959503723011581787115443755, −5.51990774945949546214482032566, −4.58480098561863554919916505543, −3.71928431960114171338843909693, −3.00798163197065683192960654202, −1.73106481391663652186129747905, −0.973948424704615278154070501327,
1.14991096354384808858007785131, 2.20338529943480945196063638614, 3.63017453870355683297213468065, 4.12920576100040620023324277408, 5.13252757496891263322980689224, 5.67125061011558069627318242177, 6.65978938754613407695935553205, 7.25464578277839164779260377080, 8.129117303197815773568401161319, 8.483928960096666577100941007817