L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s − 0.999·6-s + (0.866 + 0.5i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.5i)10-s + (−0.499 + 0.866i)12-s + 13-s + (0.866 − 0.499i)14-s − 1.73·15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + (0.499 + 0.866i)18-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s − 0.999·6-s + (0.866 + 0.5i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.5i)10-s + (−0.499 + 0.866i)12-s + 13-s + (0.866 − 0.499i)14-s − 1.73·15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + (0.499 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.442672000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.442672000\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.910208320190107449508179990731, −8.599994128198627989849980448629, −7.47071040670500424231598772457, −6.15081399121232350295169378106, −5.66084872773930760791275956933, −4.98454019402931148258479352785, −4.35027440152493127699282329617, −2.64258429747338521302425833329, −1.74929217489089042825701814728, −1.04419132045855263269346258371,
2.15352767551781304040069716982, 3.49629861808070534258901553395, 4.03310241109367512252976002514, 5.04133018439681517746332312681, 5.97776651384452038269729833118, 6.39559056397064432347120252256, 7.06576077857842648530378633544, 8.261111151983586904123363503786, 8.778782469498840694353605767027, 9.988480748877240522565842908589