L(s) = 1 | + 0.758·2-s + (−1.58 − 0.691i)3-s − 1.42·4-s − 1.15·5-s + (−1.20 − 0.524i)6-s + 4.81i·7-s − 2.59·8-s + (2.04 + 2.19i)9-s − 0.878·10-s − 4.12·11-s + (2.26 + 0.985i)12-s − 3.26i·13-s + 3.65i·14-s + (1.83 + 0.800i)15-s + 0.878·16-s + 4.51i·17-s + ⋯ |
L(s) = 1 | + 0.536·2-s + (−0.916 − 0.399i)3-s − 0.712·4-s − 0.517·5-s + (−0.491 − 0.214i)6-s + 1.82i·7-s − 0.918·8-s + (0.680 + 0.732i)9-s − 0.277·10-s − 1.24·11-s + (0.652 + 0.284i)12-s − 0.904i·13-s + 0.976i·14-s + (0.474 + 0.206i)15-s + 0.219·16-s + 1.09i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.136586 + 0.332960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.136586 + 0.332960i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.58 + 0.691i)T \) |
| 67 | \( 1 + (7.63 + 2.94i)T \) |
good | 2 | \( 1 - 0.758T + 2T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 - 4.81iT - 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 + 3.26iT - 13T^{2} \) |
| 17 | \( 1 - 4.51iT - 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 + 4.65iT - 23T^{2} \) |
| 29 | \( 1 + 2.59iT - 29T^{2} \) |
| 31 | \( 1 - 4.91iT - 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 9.17iT - 47T^{2} \) |
| 53 | \( 1 - 8.07T + 53T^{2} \) |
| 59 | \( 1 - 7.67iT - 59T^{2} \) |
| 61 | \( 1 - 1.31iT - 61T^{2} \) |
| 71 | \( 1 - 6.86iT - 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 + 6.10iT - 79T^{2} \) |
| 83 | \( 1 - 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 5.56iT - 89T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−12.50524711608610177230326927374, −12.36242831796417641370699285122, −11.08657695993031641520435962784, −10.04765952582990612182300148826, −8.592201040905909517430370567351, −7.949991968405798255149163453834, −6.06971980895927103823634528647, −5.56534451134659035455761133204, −4.49249932585622659736173039877, −2.66233779975090210601617870606,
0.29122689763761058850434738369, 3.73092337613303896400770767576, 4.42708695662352057500045337108, 5.40341400420590686579444797834, 6.87518442071879939611257946170, 7.79697662674879156461061782833, 9.408910703144388459263358340275, 10.25986868226566783189645000437, 11.16441062757485918760299267258, 12.04266690340372347347431891004