L(s) = 1 | + (−0.207 − 0.978i)2-s + (0.809 + 0.587i)3-s + (−0.913 + 0.406i)4-s + (0.406 + 0.913i)5-s + (0.406 − 0.913i)6-s + (0.866 − 0.5i)7-s + (0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (0.604 − 0.128i)11-s + (−0.978 − 0.207i)12-s + (0.336 − 1.58i)13-s + (−0.669 − 0.743i)14-s + (−0.207 + 0.978i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + ⋯ |
L(s) = 1 | + (−0.207 − 0.978i)2-s + (0.809 + 0.587i)3-s + (−0.913 + 0.406i)4-s + (0.406 + 0.913i)5-s + (0.406 − 0.913i)6-s + (0.866 − 0.5i)7-s + (0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (0.604 − 0.128i)11-s + (−0.978 − 0.207i)12-s + (0.336 − 1.58i)13-s + (−0.669 − 0.743i)14-s + (−0.207 + 0.978i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.466023929\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.466023929\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.207 + 0.978i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.406 - 0.913i)T \) |
good | 7 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.604 + 0.128i)T + (0.913 - 0.406i)T^{2} \) |
| 13 | \( 1 + (-0.336 + 1.58i)T + (-0.913 - 0.406i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.743 + 0.669i)T + (0.104 - 0.994i)T^{2} \) |
| 29 | \( 1 + (0.614 + 0.0646i)T + (0.978 + 0.207i)T^{2} \) |
| 31 | \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.614 - 0.0646i)T + (0.978 + 0.207i)T^{2} \) |
| 53 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \) |
| 67 | \( 1 + (0.169 + 1.60i)T + (-0.978 + 0.207i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.614 - 0.0646i)T + (0.978 + 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
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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
−9.476389743160849626694654446726, −8.847528558536559385020620490156, −8.011230042513075836670057087591, −7.51355834177687574173008375540, −6.15741715898720048239071093943, −5.07608228289147799284800661731, −4.10554915003561812657087186313, −3.47107567593056641111437017104, −2.52565331762483241293951477625, −1.60262148363397952770844831297,
1.37456032353390273093210821677, 2.15059946324886885792892495866, 4.00825186807989819127156865691, 4.59714891215654283931747522835, 5.56426006045675756850232638494, 6.53492086063411884762890578298, 7.09799408311822518557548096539, 8.021136566242113382315926008506, 8.824328081721517932157468982688, 9.127447055287428013815290165886