L(s) = 1 | + i·3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)7-s − 9-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)12-s + (−0.866 + 0.5i)13-s + (−0.499 + 0.866i)16-s + i·17-s + (−0.5 + 0.866i)21-s − i·27-s + 0.999i·28-s + (1 − 1.73i)29-s + (0.866 + 0.5i)33-s + (−0.5 − 0.866i)36-s + ⋯ |
L(s) = 1 | + i·3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)7-s − 9-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)12-s + (−0.866 + 0.5i)13-s + (−0.499 + 0.866i)16-s + i·17-s + (−0.5 + 0.866i)21-s − i·27-s + 0.999i·28-s + (1 − 1.73i)29-s + (0.866 + 0.5i)33-s + (−0.5 − 0.866i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.281607311\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281607311\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.861623537689002301336391450163, −8.897076713692946444073901278197, −8.374586620062652387799110053328, −7.74404023282028677716941777182, −6.51033319227279921106186720480, −5.79949499433153945011159035881, −4.67433901265314038003178145429, −4.02025632214192708359342399432, −3.01694497752613453450060376560, −2.05669390229901320017423026929,
1.07673924168719496661651400743, 1.98492320336306636557745181058, 2.98671574054719098737459256306, 4.72639573927937808472050199215, 5.19468295254110238710123648436, 6.30533529410754720474081293563, 7.19323677972223513016072040393, 7.36806660239463043594069774347, 8.480268154298305589164354944446, 9.419392947415911640140214923693