L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.173 − 0.300i)11-s + (−0.939 − 0.342i)12-s + (0.173 − 0.984i)16-s + (1.53 + 1.28i)17-s + (0.173 − 0.984i)18-s + (0.766 − 0.642i)19-s + (0.266 + 0.223i)22-s + 24-s + (0.766 − 0.642i)25-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.173 − 0.300i)11-s + (−0.939 − 0.342i)12-s + (0.173 − 0.984i)16-s + (1.53 + 1.28i)17-s + (0.173 − 0.984i)18-s + (0.766 − 0.642i)19-s + (0.266 + 0.223i)22-s + 24-s + (0.766 − 0.642i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6110586148\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6110586148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
good | 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 - 0.347T + T^{2} \) |
| 89 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
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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
−9.755394092807748162530595851818, −8.541258499626627990827897873986, −8.184163315693915419082614079336, −7.24412049464233656868783907275, −6.65260075411711102404757488422, −5.70504608035836393878162747869, −5.16511606135020667113779332743, −3.33405259242492598357121891106, −2.08700648259279201111607962034, −0.932155032862859008161388907082,
1.15927913869796972108023885925, 2.90451595872579483630719479463, 3.56587424232188166235918940197, 4.89286383570193069118792447365, 5.66506045841565804073167164308, 6.77592619028333906413875261834, 7.55977079646522791131036223693, 8.426832586535847475948917197618, 9.395465705229676889377819680891, 9.846289382140776667647137129586