L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.521 − 0.300i)5-s + (−0.499 + 0.866i)9-s + (−0.141 + 0.0818i)11-s − 3.37i·13-s + 0.601i·15-s + (1.84 − 1.06i)17-s + (−0.476 + 0.825i)19-s + (6.05 + 3.49i)23-s + (−2.31 − 4.01i)25-s + 0.999·27-s − 1.28·29-s + (−1.67 − 2.90i)31-s + (0.141 + 0.0818i)33-s + (−2.58 + 4.47i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.233 − 0.134i)5-s + (−0.166 + 0.288i)9-s + (−0.0427 + 0.0246i)11-s − 0.937i·13-s + 0.155i·15-s + (0.447 − 0.258i)17-s + (−0.109 + 0.189i)19-s + (1.26 + 0.729i)23-s + (−0.463 − 0.803i)25-s + 0.192·27-s − 0.239·29-s + (−0.301 − 0.521i)31-s + (0.0246 + 0.0142i)33-s + (−0.424 + 0.735i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9334996362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9334996362\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.521 + 0.300i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.141 - 0.0818i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 + (-1.84 + 1.06i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.476 - 0.825i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.05 - 3.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 + (1.67 + 2.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.58 - 4.47i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.67iT - 41T^{2} \) |
| 43 | \( 1 - 6.09iT - 43T^{2} \) |
| 47 | \( 1 + (-1.84 + 3.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.94 + 12.0i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.02 + 6.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.157 - 0.0907i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.87 - 4.54i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 + (-10.5 + 6.07i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.05 - 3.49i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + (12.8 + 7.44i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.08iT - 97T^{2} \) |
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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
−8.502589839105032534774682777776, −7.87338051052981623903720333152, −7.22831455639303399268168508720, −6.36281234763551724337410149044, −5.51374166366176000503449756373, −4.88906922357442083147486084242, −3.69448858859817415844306064825, −2.82389149292745278358306307615, −1.57816748932413924067744591539, −0.34695404653939249825214405539,
1.35244767761032014593947089619, 2.72026143024608748588350629262, 3.68391186840654328555165093874, 4.48266001796376388663715744078, 5.28163432882705608563317232393, 6.15235216228819519467442274801, 6.97999493859283793944605160555, 7.65177744525766192156506355535, 8.750654662799850931478956912472, 9.201046761814364545269028294393