L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−1.36 + 1.77i)5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−2.21 + 0.286i)10-s + 5.48·11-s + (0.707 − 0.707i)12-s + (2.41 + 2.41i)13-s + (2.21 − 0.286i)15-s − 1.00·16-s + (1.49 − 1.49i)17-s + (−0.707 + 0.707i)18-s − 6.99·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.610 + 0.791i)5-s − 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.701 + 0.0907i)10-s + 1.65·11-s + (0.204 − 0.204i)12-s + (0.670 + 0.670i)13-s + (0.572 − 0.0740i)15-s − 0.250·16-s + (0.363 − 0.363i)17-s + (−0.166 + 0.166i)18-s − 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.524166681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.524166681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.36 - 1.77i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 + (-2.41 - 2.41i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.49 + 1.49i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.99T + 19T^{2} \) |
| 23 | \( 1 + (1.24 - 1.24i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.684iT - 29T^{2} \) |
| 31 | \( 1 - 5.57iT - 31T^{2} \) |
| 37 | \( 1 + (-6.92 - 6.92i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.50iT - 41T^{2} \) |
| 43 | \( 1 + (1.95 - 1.95i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.49 - 2.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.64 - 6.64i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 1.17iT - 61T^{2} \) |
| 67 | \( 1 + (-7.03 - 7.03i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + (-3.44 - 3.44i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.33iT - 79T^{2} \) |
| 83 | \( 1 + (-4.05 - 4.05i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 + (-13.1 + 13.1i)T - 97iT^{2} \) |
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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
−9.732872725261521629292191952596, −8.753394279893958501035856014644, −8.039643406159433672107252865412, −7.06618291569628815365960209478, −6.45957297211122427390412566048, −6.11005726921015698466006715224, −4.59995999884151235775910461442, −4.00218679990147132759898851516, −2.99072600738344368327666561685, −1.50402197428161832805811482385,
0.57647682090715470162263885982, 1.82087959462439988349925942457, 3.53999791319369224196318761839, 4.06241205331219810777210732863, 4.76757023222270185606050193410, 5.94807821933878574139938062135, 6.37486656522709069909863518841, 7.71829416874414938954561583875, 8.654361909732466077452123889714, 9.225099504225926892681826649509