L(s) = 1 | + (0.707 − 0.707i)2-s + (0.866 − 0.5i)3-s − 1.00i·4-s + (0.0524 − 0.195i)5-s + (0.258 − 0.965i)6-s + (1.42 + 2.22i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.101 − 0.175i)10-s + (0.723 − 2.70i)11-s + (−0.500 − 0.866i)12-s + (1.90 − 3.06i)13-s + (2.58 + 0.564i)14-s + (−0.0524 − 0.195i)15-s − 1.00·16-s − 2.69·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (0.0234 − 0.0874i)5-s + (0.105 − 0.394i)6-s + (0.539 + 0.841i)7-s + (−0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.0320 − 0.0554i)10-s + (0.218 − 0.814i)11-s + (−0.144 − 0.250i)12-s + (0.528 − 0.848i)13-s + (0.690 + 0.150i)14-s + (−0.0135 − 0.0505i)15-s − 0.250·16-s − 0.652·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.92902 - 1.33004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.92902 - 1.33004i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-1.42 - 2.22i)T \) |
| 13 | \( 1 + (-1.90 + 3.06i)T \) |
good | 5 | \( 1 + (-0.0524 + 0.195i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.723 + 2.70i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + (-3.88 + 1.04i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 2.27iT - 23T^{2} \) |
| 29 | \( 1 + (-1.82 + 3.16i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.67 - 1.51i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (1.16 + 1.16i)T + 37iT^{2} \) |
| 41 | \( 1 + (10.6 - 2.85i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (5.73 - 3.31i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.83 - 1.56i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.577 + 1.00i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.48 - 3.48i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.74 - 1.00i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.87 - 2.64i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.05 - 0.283i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.07 - 11.4i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.0814 + 0.141i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.03 + 5.03i)T + 83iT^{2} \) |
| 89 | \( 1 + (10.7 - 10.7i)T - 89iT^{2} \) |
| 97 | \( 1 + (2.70 - 10.0i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.92508865004125546360057843326, −9.715925207173041964070158422520, −8.801595648579382917271757965471, −8.208748848214910113281358784233, −6.94492552143343787802632634586, −5.79068410628331811398712278223, −5.05293803679581876473058517751, −3.60381804690706030963214399227, −2.72214912776793081529192341886, −1.33799371758316384110948896725,
1.85436093728896888497997613335, 3.46640118334841142580603627443, 4.36056488127067078366918926023, 5.14098684129710754966907572216, 6.69275202160444440178552434749, 7.17341640983247419767459848350, 8.275299470215122835788203654196, 9.036917746674054908966339645441, 10.09774663346450374512611372282, 10.95263237259383578543969315962