L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.466 + 2.18i)5-s − 1.00·7-s − 1.00i·9-s + (1.89 − 1.89i)11-s + (2.65 − 2.65i)13-s + (−1.87 − 1.21i)15-s − 1.73i·17-s + (−5.33 − 5.33i)19-s + (0.707 − 0.707i)21-s − 0.160·23-s + (−4.56 + 2.04i)25-s + (0.707 + 0.707i)27-s + (−2.70 − 2.70i)29-s − 4.64·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.208 + 0.977i)5-s − 0.378·7-s − 0.333i·9-s + (0.571 − 0.571i)11-s + (0.737 − 0.737i)13-s + (−0.484 − 0.314i)15-s − 0.421i·17-s + (−1.22 − 1.22i)19-s + (0.154 − 0.154i)21-s − 0.0334·23-s + (−0.912 + 0.408i)25-s + (0.136 + 0.136i)27-s + (−0.501 − 0.501i)29-s − 0.833·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.040460207\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.040460207\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.466 - 2.18i)T \) |
good | 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 + (-1.89 + 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 + (5.33 + 5.33i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.160T + 23T^{2} \) |
| 29 | \( 1 + (2.70 + 2.70i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (7.23 + 7.23i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.79iT - 47T^{2} \) |
| 53 | \( 1 + (-3.44 - 3.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.101 - 0.101i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.01 - 6.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (-9.04 + 9.04i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.60iT - 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + (-2.04 + 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 + 3.84iT - 97T^{2} \) |
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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
−9.115088771635759570093803873773, −8.446379946409859861678400901156, −7.29738807587722399217107725524, −6.56679907197110869994227466438, −6.01452910581905769372440805641, −5.13956400967067678019928155179, −3.91112972135914303150331962826, −3.29749574937736785842401386252, −2.19257151881289387900008602761, −0.41368986701423918208319918232,
1.30578316439589386729864783007, 2.02396595507382689988927157489, 3.74381666267554764941487659508, 4.37780582112709297879744285196, 5.41299926956009347055688474269, 6.25747955712639233200704109090, 6.70869263604277332764735608130, 7.942762858996075609914457642650, 8.462682752376285921016819484797, 9.411429925232942245741408924148