L(s) = 1 | + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.500 − 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.592 + 0.342i)11-s + (0.939 − 0.342i)12-s + (−0.939 − 0.342i)16-s + (−1.11 − 1.32i)17-s + (0.173 + 0.984i)18-s + (−0.173 + 0.984i)19-s + (−0.233 + 0.642i)22-s + (0.499 − 0.866i)24-s + (0.939 − 0.342i)25-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.500 − 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.592 + 0.342i)11-s + (0.939 − 0.342i)12-s + (−0.939 − 0.342i)16-s + (−1.11 − 1.32i)17-s + (0.173 + 0.984i)18-s + (−0.173 + 0.984i)19-s + (−0.233 + 0.642i)22-s + (0.499 − 0.866i)24-s + (0.939 − 0.342i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.353084418\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353084418\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
good | 5 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \) |
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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
−11.11566401672859286280262343885, −10.46835434950025684811188695550, −9.640535587750028630255993995325, −8.869404534321264847871137462647, −7.61040188355863166937882967244, −6.32169895766462468038192638720, −5.06870835415372396495147102034, −4.47522308060303567248952059583, −3.23744707804375481069570165369, −2.26060298859621526070966466616,
2.25422590194552494922479390195, 3.36803279194637224693476486498, 4.64134433053528326880573489310, 5.90595622501318476806678510516, 6.68620586852354494227201916161, 7.52655089942636547068638560053, 8.475363609847913099215385258326, 9.022462245455943840458792314166, 10.73628345340765688341690791189, 11.52368131895017795232506185629