L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.809 + 0.587i)6-s + (0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.309 − 0.951i)13-s + (0.669 − 0.743i)16-s + (0.104 + 0.994i)17-s + (0.5 − 0.866i)18-s + (0.913 + 0.406i)19-s + (0.809 + 0.587i)22-s + (0.978 − 0.207i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (−0.809 + 0.587i)6-s + (0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.309 − 0.951i)13-s + (0.669 − 0.743i)16-s + (0.104 + 0.994i)17-s + (0.5 − 0.866i)18-s + (0.913 + 0.406i)19-s + (0.809 + 0.587i)22-s + (0.978 − 0.207i)23-s + (−0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.641925979 - 0.1028534000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.641925979 - 0.1028534000i\) |
\(L(1)\) |
\(\approx\) |
\(1.486003528 - 0.06583206205i\) |
\(L(1)\) |
\(\approx\) |
\(1.486003528 - 0.06583206205i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 11 | \( 1 + (0.669 + 0.743i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.104 + 0.994i)T \) |
| 19 | \( 1 + (0.913 + 0.406i)T \) |
| 23 | \( 1 + (0.978 - 0.207i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.669 + 0.743i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.104 - 0.994i)T \) |
| 53 | \( 1 + (-0.913 + 0.406i)T \) |
| 59 | \( 1 + (-0.978 - 0.207i)T \) |
| 61 | \( 1 + (-0.978 + 0.207i)T \) |
| 67 | \( 1 + (0.104 + 0.994i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.669 - 0.743i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.978 + 0.207i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−27.51935104396675066070414952745, −26.457552585445231388401655042532, −25.036416321750447293207061822229, −24.4166563980227217550966984350, −23.6117636640397170261963485193, −22.636603490607496813328277053727, −21.96220937040105199841613169083, −21.07181134589556703709703303167, −19.727567802842522922654481158170, −18.71804793290753731003475582755, −17.39765302736948720059637605684, −16.51536424596425258946678409378, −15.82794803410969071363103361182, −14.31239437783251168530450649602, −13.60212533796592259306903070227, −12.41407426406071652275923781043, −11.5935814205509722452807748942, −10.89434533831841411467878343090, −9.16973391762479009366382558525, −7.39236252157045687353736731548, −6.74373587060412769249469931929, −5.54928383692710207052147959423, −4.67765013937385632418007322965, −3.20953828417242849753553125061, −1.53211191281850162920610206652,
1.47122300603757253991641231215, 3.313621493621574898980398114076, 4.43747948274973468633458485985, 5.4452521546002407711352064947, 6.40301891847971339055985226781, 7.56876732135832529000932072619, 9.63197676030501020562153638679, 10.4845057103007778948171973717, 11.556923946422013662486353204474, 12.372858155754381336930824680213, 13.262207535654859462780428481174, 14.865063745278816968481971856361, 15.26457318890761100723883733449, 16.60828517198298398959235594911, 17.34448176217936082990476805518, 18.71064713890709229612190661933, 20.03967984291130138880626637858, 20.81124625651084080029648784082, 21.92933734202361751474076748851, 22.58208009691319210985111635360, 23.250607734566948944114899228647, 24.364720509797212243728455747817, 25.1656002954578352593429188395, 26.54508489200792070508186221839, 27.81159273454333117725372173838