L(s) = 1 | + i·2-s + (−1.72 + 0.146i)3-s − 4-s + 3.83·5-s + (−0.146 − 1.72i)6-s + (−0.655 + 2.56i)7-s − i·8-s + (2.95 − 0.507i)9-s + 3.83i·10-s − 3.53i·11-s + (1.72 − 0.146i)12-s + i·13-s + (−2.56 − 0.655i)14-s + (−6.61 + 0.563i)15-s + 16-s + 6.47·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.996 + 0.0848i)3-s − 0.5·4-s + 1.71·5-s + (−0.0600 − 0.704i)6-s + (−0.247 + 0.968i)7-s − 0.353i·8-s + (0.985 − 0.169i)9-s + 1.21i·10-s − 1.06i·11-s + (0.498 − 0.0424i)12-s + 0.277i·13-s + (−0.685 − 0.175i)14-s + (−1.70 + 0.145i)15-s + 0.250·16-s + 1.57·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03484 + 0.876562i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03484 + 0.876562i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.72 - 0.146i)T \) |
| 7 | \( 1 + (0.655 - 2.56i)T \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 - 3.83T + 5T^{2} \) |
| 11 | \( 1 + 3.53iT - 11T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 1.55iT - 19T^{2} \) |
| 23 | \( 1 + 3.87iT - 23T^{2} \) |
| 29 | \( 1 - 6.70iT - 29T^{2} \) |
| 31 | \( 1 - 9.76iT - 31T^{2} \) |
| 37 | \( 1 + 0.597T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.604T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 6.84iT - 53T^{2} \) |
| 59 | \( 1 - 1.74T + 59T^{2} \) |
| 61 | \( 1 - 5.28iT - 61T^{2} \) |
| 67 | \( 1 - 5.39T + 67T^{2} \) |
| 71 | \( 1 + 9.18iT - 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 - 2.06T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−10.67434023468377834338371619950, −10.15227971219411276827012018255, −9.218147940675302709522120470046, −8.534359668297378299089505035499, −6.98029885053193266867767010436, −6.17653491524693895856032011696, −5.59906927007438110467410430996, −5.07271356828813662772886116360, −3.18153150445963728054033258647, −1.43286967884311881044945785822,
1.06959639148720620644929119079, 2.21526915121517962056841104945, 3.90188702584996152548964280511, 5.11438573548834529226198986465, 5.79208473408837854839061196111, 6.82131414730190407024657751121, 7.77235238185719425982911300095, 9.573496594280620093300065589635, 9.898825009336788034376739100558, 10.35706146554417366388760131969