| L(s) = 1 | + (−1.08 − 1.34i)3-s + 5-s + (−2.53 + 0.769i)7-s + (−0.641 + 2.93i)9-s − 1.53i·11-s + 1.09i·13-s + (−1.08 − 1.34i)15-s + 1.57·17-s + 4.32i·19-s + (3.78 + 2.57i)21-s − 6.09i·23-s + 25-s + (4.65 − 2.31i)27-s − 0.867i·29-s + 4.03i·31-s + ⋯ |
| L(s) = 1 | + (−0.626 − 0.779i)3-s + 0.447·5-s + (−0.956 + 0.291i)7-s + (−0.213 + 0.976i)9-s − 0.462i·11-s + 0.302i·13-s + (−0.280 − 0.348i)15-s + 0.381·17-s + 0.992i·19-s + (0.826 + 0.562i)21-s − 1.27i·23-s + 0.200·25-s + (0.895 − 0.445i)27-s − 0.161i·29-s + 0.724i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236024299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236024299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.08 + 1.34i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.53 - 0.769i)T \) |
| good | 11 | \( 1 + 1.53iT - 11T^{2} \) |
| 13 | \( 1 - 1.09iT - 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 4.32iT - 19T^{2} \) |
| 23 | \( 1 + 6.09iT - 23T^{2} \) |
| 29 | \( 1 + 0.867iT - 29T^{2} \) |
| 31 | \( 1 - 4.03iT - 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.51iT - 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 3.69T + 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 2.71iT - 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 - 6.68T + 83T^{2} \) |
| 89 | \( 1 + 4.13T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−9.256301945614319272570455621084, −8.457197681737161626373474321842, −7.59457372906540162857201208562, −6.66999354164362559802279781319, −6.09599881009560452755648955527, −5.53355021939278712135459272038, −4.35529887909960174059320733332, −3.07454905151735214430194626922, −2.12330793705051119918805761847, −0.76978859066532871414294333515,
0.812684223410634466936775346280, 2.60017931275434419564797730464, 3.58695221748742815536589009596, 4.45386602807734933164847152011, 5.43140991260963380000306672247, 6.06627483853619244366556239203, 6.89195812063018316261049680758, 7.71795875863128272041431153424, 9.060250870469811205574543905592, 9.535262812770751366326608244719
