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
−31.42419950625557560991030902424, −30.2391149190694466077968694706, −29.01267597237802035788253517126, −28.35801267427729113521190655322, −27.10199320863722619690751530118, −26.25391128660214742663111631951, −25.701118457208868194290113099007, −24.18882937147961157412626360937, −22.39957849752455733144768041223, −21.599965217010592177804059128170, −20.643822960809841932325042757904, −19.44413321996789631775185117384, −18.13955308679429178660593983553, −17.27024389270937391982824799106, −16.10847936282743548538623755600, −15.10383346502125011903092665089, −13.72353871669230264298217521483, −11.636421085167728843044671405497, −10.82159132447325427472777142582, −9.7072349491986737477046949939, −8.90566240860414258671071025466, −7.10214338142543931851792881622, −5.790514889983247207780073431035, −3.72115519101261125281382675087, −2.2232882267185419524740772252,
1.05579301589600281019069120816, 2.47472211027939259189002604237, 5.4481447631619617913231806570, 6.50409605319997825251682704990, 7.91337423597837233491926456012, 8.84584975415468956753847568504, 10.224605654782553723421533785598, 11.66547160483855585945065247982, 12.8556229474564342447183662949, 14.05891444639369134962844692371, 15.72335685613782531630199182112, 16.959757231466721125116036703821, 17.789116798187120151436878795, 18.59661398109408630387540607295, 20.00897997198843423262442436068, 20.61165175582164104614533120, 22.46600825055167831486939390296, 23.96083523785999049294583319083, 24.739385632662065230234680828957, 25.38730309718046578269528442829, 26.62721483633812466652959319854, 28.08012096265825180089638759401, 28.777815273627585115605938447993, 29.62669086965922331256430140957, 30.67659348172208514293503444924