L(s) = 1 | + (1.43 − 0.524i)2-s + (−0.173 + 0.984i)3-s + (1.03 − 0.866i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 1.50i)6-s + (0.266 − 0.460i)8-s + (−0.939 − 0.342i)9-s + (−1.43 − 0.524i)10-s + (0.673 + 1.16i)12-s + (0.766 − 0.642i)15-s + (−0.0923 + 0.524i)16-s + (−1.76 + 0.642i)17-s − 1.53·18-s + (0.173 − 0.984i)19-s − 1.34·20-s + ⋯ |
L(s) = 1 | + (1.43 − 0.524i)2-s + (−0.173 + 0.984i)3-s + (1.03 − 0.866i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 1.50i)6-s + (0.266 − 0.460i)8-s + (−0.939 − 0.342i)9-s + (−1.43 − 0.524i)10-s + (0.673 + 1.16i)12-s + (0.766 − 0.642i)15-s + (−0.0923 + 0.524i)16-s + (−1.76 + 0.642i)17-s − 1.53·18-s + (0.173 − 0.984i)19-s − 1.34·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.254450995\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.254450995\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.173 + 0.984i)T \) |
good | 2 | \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
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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
−12.04974984040644359312944856215, −11.17055883961491829220116965573, −10.77331013510667715622969303243, −9.189338096976710823030278747713, −8.485520438667346134609467973120, −6.74414085738013175794918733852, −5.49348741541786826041408957845, −4.54861483542438634931694193169, −4.04411960311907984571054516076, −2.71484977217962682589049111425,
2.60031055058779045732555514958, 3.81251638633129187039886799262, 5.06412496625005352757204882521, 6.20547253459175653183572755737, 6.99288876793721282594679364652, 7.62084970447560224560039398688, 8.921763971370198173318857569383, 10.71387020893963286388797957291, 11.63055144168683879737137454858, 12.17056755729535196012655792889