L(s) = 1 | + 1.27·2-s + (1.36 − 1.06i)3-s − 0.363·4-s + 0.284·5-s + (1.75 − 1.35i)6-s + 3.33i·7-s − 3.02·8-s + (0.752 − 2.90i)9-s + 0.363·10-s − 3.20i·11-s + (−0.497 + 0.385i)12-s − 2.50·13-s + 4.26i·14-s + (0.389 − 0.301i)15-s − 3.14·16-s + 1.65i·17-s + ⋯ |
L(s) = 1 | + 0.904·2-s + (0.790 − 0.612i)3-s − 0.181·4-s + 0.127·5-s + (0.715 − 0.553i)6-s + 1.25i·7-s − 1.06·8-s + (0.250 − 0.968i)9-s + 0.114·10-s − 0.966i·11-s + (−0.143 + 0.111i)12-s − 0.694·13-s + 1.13i·14-s + (0.100 − 0.0777i)15-s − 0.785·16-s + 0.401i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72395 - 0.280478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72395 - 0.280478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.36 + 1.06i)T \) |
| 43 | \( 1 + (3.64 + 5.45i)T \) |
good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 5 | \( 1 - 0.284T + 5T^{2} \) |
| 7 | \( 1 - 3.33iT - 7T^{2} \) |
| 11 | \( 1 + 3.20iT - 11T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 - 1.65iT - 17T^{2} \) |
| 19 | \( 1 - 6.22iT - 19T^{2} \) |
| 23 | \( 1 + 1.05iT - 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 7.64T + 31T^{2} \) |
| 37 | \( 1 + 0.770iT - 37T^{2} \) |
| 41 | \( 1 + 4.36iT - 41T^{2} \) |
| 47 | \( 1 + 12.7iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 5.80iT - 59T^{2} \) |
| 61 | \( 1 - 8.78iT - 61T^{2} \) |
| 67 | \( 1 - 2.14T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 5.31iT - 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 - 9.01iT - 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 2.45T + 97T^{2} \) |
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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
−13.46801358769813159357185597875, −12.17458531964022595014598036684, −12.07961759570969091041088917974, −9.968125121661696998912079350169, −8.824546459019844255007139734543, −8.149983309336682831456876258528, −6.35273620166791776606645973639, −5.48516400037804152942941117750, −3.75639711475773821110208409319, −2.48612534692354645278597477086,
2.86167885188078477429493673098, 4.31764763788153682476247219711, 4.86030187911481469174773834800, 6.82066670207291034113153001930, 8.002472563609429703945100799871, 9.475504369637006476729312958136, 10.01277951682933933726093729197, 11.42756274659242347794572083500, 12.78553999832992765004007118274, 13.61460426224889882569791593196