L(s) = 1 | + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.156 + 0.987i)5-s + (−1.26 + 1.26i)7-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)10-s + (−1.16 − 1.59i)11-s + (1.44 − 1.04i)14-s + (0.809 + 0.587i)16-s + (−0.453 + 0.891i)20-s + (0.896 + 1.76i)22-s + (−0.951 − 0.309i)25-s + (−1.58 + 0.809i)28-s + (−0.437 + 1.34i)29-s + (−0.587 − 1.80i)31-s + (−0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.156 + 0.987i)5-s + (−1.26 + 1.26i)7-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)10-s + (−1.16 − 1.59i)11-s + (1.44 − 1.04i)14-s + (0.809 + 0.587i)16-s + (−0.453 + 0.891i)20-s + (0.896 + 1.76i)22-s + (−0.951 − 0.309i)25-s + (−1.58 + 0.809i)28-s + (−0.437 + 1.34i)29-s + (−0.587 − 1.80i)31-s + (−0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.008754437862\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.008754437862\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.156 - 0.987i)T \) |
good | 7 | \( 1 + (1.26 - 1.26i)T - iT^{2} \) |
| 11 | \( 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.253 - 0.183i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.550 + 0.280i)T + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \) |
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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
−9.132884664430485985067361830159, −8.485937410925689267199773657012, −7.69508468744806378678002037072, −6.84305946488816248947525670871, −5.96829857430217243638907080309, −5.64966853864704235409588798072, −3.50547463994565677586348641066, −3.00492956398023495268618850838, −2.26596409322612176906348974025, −0.008554795011706307577689978402,
1.47994800551165332851957846553, 2.76754724568169578823778752196, 3.99274673574568988243870348259, 4.93509476241500222471179710987, 5.94042198378166192970032881371, 7.02001696738273066511345310831, 7.39418140394358267824701489443, 8.158745820909820608320733554552, 9.142043159353979060852439516576, 9.822543587830719140679850685102