L(s) = 1 | − 2-s + (1.68 + 0.420i)3-s + 4-s − 0.841i·5-s + (−1.68 − 0.420i)6-s + (−0.595 − 2.57i)7-s − 8-s + (2.64 + 1.41i)9-s + 0.841i·10-s + (1.68 + 0.420i)12-s + (2.27 − 2.79i)13-s + (0.595 + 2.57i)14-s + (0.354 − 1.41i)15-s + 16-s − 4.33·17-s + (−2.64 − 1.41i)18-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.970 + 0.242i)3-s + 0.5·4-s − 0.376i·5-s + (−0.685 − 0.171i)6-s + (−0.224 − 0.974i)7-s − 0.353·8-s + (0.881 + 0.471i)9-s + 0.266i·10-s + (0.485 + 0.121i)12-s + (0.631 − 0.775i)13-s + (0.159 + 0.688i)14-s + (0.0914 − 0.365i)15-s + 0.250·16-s − 1.05·17-s + (−0.623 − 0.333i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36601 - 0.499767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36601 - 0.499767i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.68 - 0.420i)T \) |
| 7 | \( 1 + (0.595 + 2.57i)T \) |
| 13 | \( 1 + (-2.27 + 2.79i)T \) |
good | 5 | \( 1 + 0.841iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 7.98iT - 23T^{2} \) |
| 29 | \( 1 - 2.32iT - 29T^{2} \) |
| 31 | \( 1 + 5.53T + 31T^{2} \) |
| 37 | \( 1 + 5.15iT - 37T^{2} \) |
| 41 | \( 1 - 10.8iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 7.82iT - 47T^{2} \) |
| 53 | \( 1 + 7.98iT - 53T^{2} \) |
| 59 | \( 1 + 3.91iT - 59T^{2} \) |
| 61 | \( 1 - 5.59iT - 61T^{2} \) |
| 67 | \( 1 + 1.82iT - 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 7.27iT - 83T^{2} \) |
| 89 | \( 1 + 12.5iT - 89T^{2} \) |
| 97 | \( 1 + 9.87T + 97T^{2} \) |
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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
−10.61994836100126971689446484997, −9.675568786642099189402981126594, −8.936307152643665593850537068589, −8.176128757488158534233722227742, −7.36846486942716592690031555071, −6.46323125117243766617005994359, −4.88252589297135351334398642731, −3.78491915492665488540176545446, −2.67186854168807657767385234989, −1.05062880503702749345095537488,
1.69696089241618533584007584827, 2.76493024661493328844778085886, 3.83278288394319575204873458359, 5.54221846101711792989604356115, 6.70473478008951854840208275539, 7.36767097918541731250715784720, 8.455502276760436065798929161991, 9.149404292141893388595623262291, 9.597601828003980588620624603407, 10.84831466490237847557465126819