L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 1.41·6-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (−0.5 + 0.866i)12-s − 0.999·15-s + (0.499 − 0.866i)16-s + (0.707 + 1.22i)18-s + (−0.707 + 1.22i)19-s − 0.999·20-s + (−0.707 + 1.22i)23-s + (−0.499 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 1.41·6-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (−0.5 + 0.866i)12-s − 0.999·15-s + (0.499 − 0.866i)16-s + (0.707 + 1.22i)18-s + (−0.707 + 1.22i)19-s − 0.999·20-s + (−0.707 + 1.22i)23-s + (−0.499 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.258446815\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.258446815\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.40568904514326495869467866212, −9.756814152118602413413838495718, −8.480430225670073224715954229413, −7.71546850894808407275465067059, −6.41694160162284171093223295223, −5.51504386370411204034742997640, −4.76418900317356564866494229436, −3.58242554572332453256505996505, −2.15150059135229144708006282839, −1.37810325116112342368004411780,
2.63487817523132710027597905007, 4.02943667866464755173379376161, 4.74432837941686459881135014539, 5.81323282971528958859886389574, 6.34927474447112696564101843963, 7.06892040177764118705906923877, 8.216515549202172755489527326506, 9.246615387143634888898785177788, 10.23859687610117494909045191519, 10.78558210702618647456507014128