L(s) = 1 | + (−0.978 − 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (0.309 + 0.951i)20-s + ⋯ |
L(s) = 1 | + (−0.978 − 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (0.669 + 0.743i)5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + (0.309 + 0.951i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0395 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0395 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4589780525 + 0.4411525607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4589780525 + 0.4411525607i\) |
\(L(1)\) |
\(\approx\) |
\(0.6476723973 + 0.2985769157i\) |
\(L(1)\) |
\(\approx\) |
\(0.6476723973 + 0.2985769157i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (0.669 + 0.743i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + (-0.104 - 0.994i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.913 - 0.406i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.913 + 0.406i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−30.67659348172208514293503444924, −29.62669086965922331256430140957, −28.777815273627585115605938447993, −28.08012096265825180089638759401, −26.62721483633812466652959319854, −25.38730309718046578269528442829, −24.739385632662065230234680828957, −23.96083523785999049294583319083, −22.46600825055167831486939390296, −20.61165175582164104614533120, −20.00897997198843423262442436068, −18.59661398109408630387540607295, −17.789116798187120151436878795, −16.959757231466721125116036703821, −15.72335685613782531630199182112, −14.05891444639369134962844692371, −12.8556229474564342447183662949, −11.66547160483855585945065247982, −10.224605654782553723421533785598, −8.84584975415468956753847568504, −7.91337423597837233491926456012, −6.50409605319997825251682704990, −5.4481447631619617913231806570, −2.47472211027939259189002604237, −1.05579301589600281019069120816,
2.2232882267185419524740772252, 3.72115519101261125281382675087, 5.790514889983247207780073431035, 7.10214338142543931851792881622, 8.90566240860414258671071025466, 9.7072349491986737477046949939, 10.82159132447325427472777142582, 11.636421085167728843044671405497, 13.72353871669230264298217521483, 15.10383346502125011903092665089, 16.10847936282743548538623755600, 17.27024389270937391982824799106, 18.13955308679429178660593983553, 19.44413321996789631775185117384, 20.643822960809841932325042757904, 21.599965217010592177804059128170, 22.39957849752455733144768041223, 24.18882937147961157412626360937, 25.701118457208868194290113099007, 26.25391128660214742663111631951, 27.10199320863722619690751530118, 28.35801267427729113521190655322, 29.01267597237802035788253517126, 30.2391149190694466077968694706, 31.42419950625557560991030902424