L(s) = 1 | + (0.5 − 0.866i)2-s + 3.24·3-s + (−0.499 − 0.866i)4-s + 5-s + (1.62 − 2.81i)6-s + (−1.12 − 1.94i)7-s − 0.999·8-s + 7.55·9-s + (0.5 − 0.866i)10-s + (−2 − 3.46i)11-s + (−1.62 − 2.81i)12-s + (0.470 − 0.815i)13-s − 2.24·14-s + 3.24·15-s + (−0.5 + 0.866i)16-s + (−2.65 + 4.59i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + 1.87·3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.663 − 1.14i)6-s + (−0.425 − 0.736i)7-s − 0.353·8-s + 2.51·9-s + (0.158 − 0.273i)10-s + (−0.603 − 1.04i)11-s + (−0.468 − 0.812i)12-s + (0.130 − 0.226i)13-s − 0.601·14-s + 0.838·15-s + (−0.125 + 0.216i)16-s + (−0.643 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.64483 - 1.81141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.64483 - 1.81141i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (7.28 - 3.73i)T \) |
good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 7 | \( 1 + (1.12 + 1.94i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.470 + 0.815i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.65 - 4.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.30 - 7.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.124 - 0.215i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0586 - 0.101i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.09 - 7.09i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.21 + 5.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.27 + 3.94i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-6.12 - 10.6i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.175T + 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 + (-5.15 + 8.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-2.34 - 4.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.90 - 5.02i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.412 - 0.713i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.12 + 8.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + (6.55 - 11.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.40652448463416181453914935526, −9.551809864393625363649489717825, −8.479760086723009320990965176949, −8.201491090440185241578771532332, −6.88836598115259082066625229847, −5.82264551367987752027364724860, −4.19775664039105472473519965056, −3.59549658250935952564045527986, −2.64916966730529240752113311068, −1.54442845858244255029643721241,
2.38600852777455654129628756125, 2.68261075957153917046498885239, 4.20858938532067516266792255551, 4.97706417389587619339615081198, 6.56214875070945132132297286356, 7.15689618450410337398788721895, 8.148445836386590807512788937646, 8.951475637610369013959891923982, 9.416472248195583654915408370315, 10.21543464406560023811757466077