L(s) = 1 | + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + 2.44i·6-s + (1.88 + 6.74i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (1.65 + 2.87i)11-s + (−2.99 − 1.73i)12-s − 19.5i·13-s + (−9.58 − 2.45i)14-s + (−2.00 + 3.46i)16-s + (7.30 − 4.21i)17-s + (2.12 + 3.67i)18-s + (0.704 + 0.406i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + 0.408i·6-s + (0.269 + 0.963i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.150 + 0.260i)11-s + (−0.249 − 0.144i)12-s − 1.50i·13-s + (−0.684 − 0.175i)14-s + (−0.125 + 0.216i)16-s + (0.429 − 0.247i)17-s + (0.117 + 0.204i)18-s + (0.0370 + 0.0214i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.956990062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.956990062\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.88 - 6.74i)T \) |
good | 11 | \( 1 + (-1.65 - 2.87i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 19.5iT - 169T^{2} \) |
| 17 | \( 1 + (-7.30 + 4.21i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.704 - 0.406i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-5.98 + 10.3i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 4.68T + 841T^{2} \) |
| 31 | \( 1 + (-27.9 + 16.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (3.01 - 5.21i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 19.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.53T + 1.84e3T^{2} \) |
| 47 | \( 1 + (15.8 + 9.14i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-15.9 - 27.6i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-64.3 + 37.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-95.1 - 54.9i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (47.3 + 81.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 98.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (9.62 - 5.55i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (0.500 - 0.867i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 66.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-133. - 77.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 132. iT - 9.40e3T^{2} \) |
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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
−9.634118055395700218699094486789, −8.648705918042430707844353706027, −8.167644337891674461917179090674, −7.42074669786918546511907152801, −6.39271114190961282633153911112, −5.57970647741992267670193405544, −4.73737034316792765503527143499, −3.27224230888561936775132405182, −2.26674156052718048849022483168, −0.815077286626769900811265313588,
1.04291404735233927872442745669, 2.14576997815723873133907546934, 3.47542074162591612023334386315, 4.13272648286184821953386876960, 5.06467630549352596287267299014, 6.57487899036751435246635779854, 7.33111932118090742509632521923, 8.258295714430605015431759980915, 8.943419938844163724765396187436, 9.797960517149978646842037941821