| L(s) = 1 | + 3-s + 2.89i·5-s − i·7-s + 9-s − 4.62i·11-s + (−2.28 − 2.78i)13-s + 2.89i·15-s + 2.68·17-s − 6.97i·19-s − i·21-s + 5.74·23-s − 3.40·25-s + 27-s − 9.63·29-s − 4.95i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.29i·5-s − 0.377i·7-s + 0.333·9-s − 1.39i·11-s + (−0.633 − 0.773i)13-s + 0.748i·15-s + 0.650·17-s − 1.59i·19-s − 0.218i·21-s + 1.19·23-s − 0.681·25-s + 0.192·27-s − 1.78·29-s − 0.889i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.935733108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935733108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (2.28 + 2.78i)T \) |
| good | 5 | \( 1 - 2.89iT - 5T^{2} \) |
| 11 | \( 1 + 4.62iT - 11T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 + 6.97iT - 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 + 9.63T + 29T^{2} \) |
| 31 | \( 1 + 4.95iT - 31T^{2} \) |
| 37 | \( 1 - 3.11iT - 37T^{2} \) |
| 41 | \( 1 + 9.43iT - 41T^{2} \) |
| 43 | \( 1 + 2.54T + 43T^{2} \) |
| 47 | \( 1 - 9.87iT - 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 + 5.56iT - 67T^{2} \) |
| 71 | \( 1 + 5.50iT - 71T^{2} \) |
| 73 | \( 1 - 6.20iT - 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 15.9iT - 83T^{2} \) |
| 89 | \( 1 - 2.24iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−9.034957927972840788841236895473, −8.040661246081444195713630404921, −7.38160005054868436865090832970, −6.83442004833136511889868655523, −5.86331855267026028572686301675, −5.00164522928336454057960249200, −3.65516833774808805316009442036, −3.12367545678894231187811401067, −2.38718344379875156605858276074, −0.63984112586705204329937141577,
1.40579565133565849782169082553, 2.10269585014341619954462958569, 3.46985010378613250653875123066, 4.42291032738615338199478029509, 5.04097029723749753784533927659, 5.85493403874099952372129128429, 7.18526440698251712958697314251, 7.55520678866488583867731818494, 8.645510418347036958514030283663, 9.022644695140520272931962148019
