L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.587 − 0.809i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + 18-s + (−0.951 + 0.309i)22-s − 1.90i·23-s + (0.951 − 0.309i)25-s + i·28-s + (1.76 − 0.896i)29-s − i·32-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.587 − 0.809i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + 18-s + (−0.951 + 0.309i)22-s − 1.90i·23-s + (0.951 − 0.309i)25-s + i·28-s + (1.76 − 0.896i)29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6489117398\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6489117398\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 7 | \( 1 + (0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 5 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 23 | \( 1 + 1.90iT - T^{2} \) |
| 29 | \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.221 - 0.221i)T - iT^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 61 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 67 | \( 1 + (0.221 + 0.221i)T + iT^{2} \) |
| 71 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−9.910494402572442928689207562611, −8.820120175295947136700313967289, −8.545654979028674289088485736958, −7.21759418210700034722806115839, −6.51319406833055909848742847382, −6.19907445349351421707570578801, −4.84391607162369303681316926007, −3.94395889201690143792304765044, −2.59003425618363610365452595590, −0.76189047201252158478853776405,
1.40624807326241964942146145526, 2.87836008686890698794943599836, 3.37816600255881236383048990798, 4.67842571790251283856538110337, 5.76034763182165160204358508638, 6.79353521536317118168759590536, 7.65370052115270694404476893120, 8.756720395114852532110997391642, 8.985508222394240747796632257368, 10.02190765106153097263614331153