L(s) = 1 | + (0.0922 − 0.995i)2-s + (0.445 + 0.895i)3-s + (−0.982 − 0.183i)4-s + (−0.602 + 0.798i)5-s + (0.932 − 0.361i)6-s + (−0.273 − 0.961i)7-s + (−0.273 + 0.961i)8-s + (−0.602 + 0.798i)9-s + (0.739 + 0.673i)10-s + (−0.602 − 0.798i)11-s + (−0.273 − 0.961i)12-s + (−0.273 − 0.961i)13-s + (−0.982 + 0.183i)14-s + (−0.982 − 0.183i)15-s + (0.932 + 0.361i)16-s + (0.445 + 0.895i)17-s + ⋯ |
L(s) = 1 | + (0.0922 − 0.995i)2-s + (0.445 + 0.895i)3-s + (−0.982 − 0.183i)4-s + (−0.602 + 0.798i)5-s + (0.932 − 0.361i)6-s + (−0.273 − 0.961i)7-s + (−0.273 + 0.961i)8-s + (−0.602 + 0.798i)9-s + (0.739 + 0.673i)10-s + (−0.602 − 0.798i)11-s + (−0.273 − 0.961i)12-s + (−0.273 − 0.961i)13-s + (−0.982 + 0.183i)14-s + (−0.982 − 0.183i)15-s + (0.932 + 0.361i)16-s + (0.445 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.079133030 - 0.4738453248i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.079133030 - 0.4738453248i\) |
\(L(1)\) |
\(\approx\) |
\(0.9602132841 - 0.2427995486i\) |
\(L(1)\) |
\(\approx\) |
\(0.9602132841 - 0.2427995486i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 647 | \( 1 \) |
good | 2 | \( 1 + (0.0922 - 0.995i)T \) |
| 3 | \( 1 + (0.445 + 0.895i)T \) |
| 5 | \( 1 + (-0.602 + 0.798i)T \) |
| 7 | \( 1 + (-0.273 - 0.961i)T \) |
| 11 | \( 1 + (-0.602 - 0.798i)T \) |
| 13 | \( 1 + (-0.273 - 0.961i)T \) |
| 17 | \( 1 + (0.445 + 0.895i)T \) |
| 19 | \( 1 + (0.932 + 0.361i)T \) |
| 23 | \( 1 + (0.932 - 0.361i)T \) |
| 29 | \( 1 + (0.932 - 0.361i)T \) |
| 31 | \( 1 + (-0.602 - 0.798i)T \) |
| 37 | \( 1 + (0.932 - 0.361i)T \) |
| 41 | \( 1 + (-0.273 + 0.961i)T \) |
| 43 | \( 1 + (0.0922 + 0.995i)T \) |
| 47 | \( 1 + (0.445 + 0.895i)T \) |
| 53 | \( 1 + (0.932 - 0.361i)T \) |
| 59 | \( 1 + (0.739 + 0.673i)T \) |
| 61 | \( 1 + (0.445 - 0.895i)T \) |
| 67 | \( 1 + (0.445 - 0.895i)T \) |
| 71 | \( 1 + (0.932 - 0.361i)T \) |
| 73 | \( 1 + (0.0922 - 0.995i)T \) |
| 79 | \( 1 + (0.0922 + 0.995i)T \) |
| 83 | \( 1 + (0.739 - 0.673i)T \) |
| 89 | \( 1 + (0.0922 - 0.995i)T \) |
| 97 | \( 1 + (0.739 + 0.673i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.498776040175251623937799249957, −22.45739470899957978382359712609, −21.39093099691859746033853160559, −20.44337092358608064016150147678, −19.489896534602197653601210032108, −18.6999344952960425475280921520, −18.12758220959998586213840440614, −17.18386902103226649463087001906, −16.19155585259453740247855634588, −15.5561858809720539112347416925, −14.756577175589542060663471015012, −13.81433354190546041040519322080, −13.01134186607887983198483712228, −12.23445101520150206312316748786, −11.75240485201239064646254087561, −9.686777478467923275358299656, −9.01541086273432671494865663693, −8.39918925038692263158299455398, −7.25291587192985242423180805428, −6.99272189956624265820706301643, −5.47674271829200332350111967726, −4.97360333621720316324219356231, −3.60654481584308096731646065442, −2.45999068908803718116748379812, −0.9454861460713770536787276105,
0.73957997239233705588816265844, 2.64129438779228519979209005606, 3.26438660431886569277617090915, 3.90200011644413750002307011254, 4.93266763641699864463028975274, 6.033117136847019738565729565949, 7.766666285010033805626412271326, 8.133280896079664360635105445282, 9.51978095009309633547386158297, 10.28542482263714980107763227421, 10.773045119836994170960104231654, 11.45220923211132612534770118847, 12.7745789015342337937719979797, 13.56620870401325600602435276363, 14.43755145395691217145031789004, 15.0324734213759676173892713312, 16.10685352428039919598865389978, 16.96002089865195808733955733860, 18.06577137697411636492299403799, 18.994804399857887609557586647421, 19.67592362301413261226913421926, 20.235269977798348325291489597614, 21.11894758400117512113446012605, 21.815293191641030685212876939253, 22.751730791282841638304267710098