L(s) = 1 | + (1.40 − 0.189i)2-s + (1.92 − 0.532i)4-s + 1.46i·5-s + (−2.07 + 1.63i)7-s + (2.60 − 1.11i)8-s + (0.278 + 2.05i)10-s + 4.88i·11-s + 6.31i·13-s + (−2.60 + 2.69i)14-s + (3.43 − 2.05i)16-s − 4.18i·17-s − 0.299·19-s + (0.779 + 2.82i)20-s + (0.927 + 6.84i)22-s − 4.88i·23-s + ⋯ |
L(s) = 1 | + (0.990 − 0.134i)2-s + (0.963 − 0.266i)4-s + 0.654i·5-s + (−0.785 + 0.619i)7-s + (0.919 − 0.393i)8-s + (0.0879 + 0.648i)10-s + 1.47i·11-s + 1.75i·13-s + (−0.695 + 0.718i)14-s + (0.858 − 0.513i)16-s − 1.01i·17-s − 0.0687·19-s + (0.174 + 0.631i)20-s + (0.197 + 1.46i)22-s − 1.01i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.34348 + 1.18593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.34348 + 1.18593i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 + 0.189i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.07 - 1.63i)T \) |
good | 5 | \( 1 - 1.46iT - 5T^{2} \) |
| 11 | \( 1 - 4.88iT - 11T^{2} \) |
| 13 | \( 1 - 6.31iT - 13T^{2} \) |
| 17 | \( 1 + 4.18iT - 17T^{2} \) |
| 19 | \( 1 + 0.299T + 19T^{2} \) |
| 23 | \( 1 + 4.88iT - 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 4.62iT - 41T^{2} \) |
| 43 | \( 1 + 5.16iT - 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 5.40iT - 61T^{2} \) |
| 67 | \( 1 - 6.31iT - 67T^{2} \) |
| 71 | \( 1 + 6.13iT - 71T^{2} \) |
| 73 | \( 1 - 1.89iT - 73T^{2} \) |
| 79 | \( 1 - 14.5iT - 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 5.40iT - 97T^{2} \) |
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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
−10.57063974297983955699540550845, −9.727756422812208647699365480908, −9.020887042005766294172265939550, −7.36339069707798427295293157720, −6.83797434562688015034740031532, −6.17372955045596798416680478882, −4.87884041031581085494247604315, −4.14344414706906997471681875537, −2.82241614078852402280602057605, −2.07322209119362147287437960184,
1.01741365877908972322839529865, 3.02187558296250523529916435628, 3.59720874707717610655733578597, 4.81563039945635328248291813235, 5.85370703300589338157318316967, 6.28831830936653235294411570961, 7.70939000641447732328525165004, 8.173485957677772707899999027971, 9.402856972715863029689979513297, 10.58444767323362202491152685304