L(s) = 1 | + (1.22 − 0.707i)2-s + (−0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s + 2.44i·6-s + (4.79 − 5.10i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (0.919 − 1.59i)11-s + (1.73 + 2.99i)12-s − 5.40·13-s + (2.26 − 9.63i)14-s + (−2.00 − 3.46i)16-s + (5.02 − 8.71i)17-s + (−3.67 − 2.12i)18-s + (7.96 − 4.59i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 + 0.5i)3-s + (0.249 − 0.433i)4-s + 0.408i·6-s + (0.684 − 0.728i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.0835 − 0.144i)11-s + (0.144 + 0.249i)12-s − 0.415·13-s + (0.161 − 0.688i)14-s + (−0.125 − 0.216i)16-s + (0.295 − 0.512i)17-s + (−0.204 − 0.117i)18-s + (0.418 − 0.241i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.222092133\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.222092133\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-4.79 + 5.10i)T \) |
good | 11 | \( 1 + (-0.919 + 1.59i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 5.40T + 169T^{2} \) |
| 17 | \( 1 + (-5.02 + 8.71i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-7.96 + 4.59i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.797 - 0.460i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 12.5T + 841T^{2} \) |
| 31 | \( 1 + (36.1 + 20.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (6.30 - 3.64i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.12iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (16.9 + 29.4i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-90.1 - 52.0i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (12.5 + 7.26i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.5 + 11.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (12.9 + 7.46i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 17.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-62.1 + 107. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (40.4 + 70.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 154.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (58.9 - 34.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 88.1T + 9.40e3T^{2} \) |
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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
−9.696710909654216170481768278333, −8.852934382228278153093874396520, −7.61942458623690878078580124399, −6.99562610122377526810194869178, −5.72753425205615516383829347534, −5.07688051410025593036455370012, −4.18984683898627425043101558290, −3.37513200451126292711090587309, −2.01048052946302395689694456080, −0.57731393311130869287391477543,
1.49882627173043482135759009864, 2.57660938750015074823483613199, 3.83522389552855108065468778005, 5.03112669091027154746644802228, 5.56273014005323528809290220914, 6.49138994327566808993882497025, 7.39811847571784761056646838782, 8.087579377716917566453717757437, 8.913593292617083400369872323567, 9.966969932608653290597758791335