| L(s) = 1 | + 1.66i·2-s − 0.756·4-s + 5.20·7-s + 2.06i·8-s − 4.92i·11-s − 2.72·13-s + 8.64i·14-s − 4.94·16-s + 8.24i·17-s + 8.17·22-s − 5·25-s − 4.52i·26-s − 3.94·28-s − 5.38i·29-s − 4.07i·32-s + ⋯ |
| L(s) = 1 | + 1.17i·2-s − 0.378·4-s + 1.96·7-s + 0.729i·8-s − 1.48i·11-s − 0.755·13-s + 2.31i·14-s − 1.23·16-s + 1.99i·17-s + 1.74·22-s − 25-s − 0.886i·26-s − 0.744·28-s − 0.999i·29-s − 0.720i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07484 + 1.07484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07484 + 1.07484i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + 5.38iT \) |
| good | 2 | \( 1 - 1.66iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 + 4.92iT - 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 - 8.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 10.7iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 1.71iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 5.03iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.03652163979263385801882432372, −11.24822468560849133378808567378, −10.52289335232629618531489032354, −8.726773607157863540082459929124, −8.176521000502004765071255331715, −7.51309383531782208911058307717, −6.06170509694916636370863239081, −5.40524839820306090859703475933, −4.15989143815161087958658312826, −1.98572221001153328727913236702,
1.56883344954744370225912333409, 2.59583149722910636997990759134, 4.44485243912479532962472500168, 5.03589232272127142242340766968, 7.09906025136613125177566008306, 7.74572370966521745757064712326, 9.234066192340207170951113841802, 9.998474436792621228958608904959, 11.01758936567099582910865255609, 11.77960916034356913783494753256
