| L(s) = 1 | − 8.10·2-s + 9.59·3-s + 33.6·4-s + 25·5-s − 77.7·6-s − 225.·7-s − 13.5·8-s − 150.·9-s − 202.·10-s − 739.·11-s + 323.·12-s + 220.·13-s + 1.82e3·14-s + 239.·15-s − 967.·16-s + 899.·17-s + 1.22e3·18-s + 407.·19-s + 841.·20-s − 2.15e3·21-s + 5.99e3·22-s − 402.·23-s − 129.·24-s + 625·25-s − 1.78e3·26-s − 3.77e3·27-s − 7.58e3·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 0.615·3-s + 1.05·4-s + 0.447·5-s − 0.881·6-s − 1.73·7-s − 0.0747·8-s − 0.621·9-s − 0.640·10-s − 1.84·11-s + 0.647·12-s + 0.362·13-s + 2.48·14-s + 0.275·15-s − 0.945·16-s + 0.754·17-s + 0.889·18-s + 0.259·19-s + 0.470·20-s − 1.06·21-s + 2.64·22-s − 0.158·23-s − 0.0459·24-s + 0.200·25-s − 0.518·26-s − 0.997·27-s − 1.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5729585369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5729585369\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 + 8.10T + 32T^{2} \) |
| 3 | \( 1 - 9.59T + 243T^{2} \) |
| 7 | \( 1 + 225.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 739.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 220.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 899.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 407.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 402.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 649.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.35e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.10e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.82e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.09e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.27e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.01e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.10e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.28e3T + 8.58e9T^{2} \) |
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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
−10.91522360646167686803765383689, −10.06873506978186275250001834568, −9.572940781948926011327783882000, −8.560024094341833702830684423424, −7.81194221508058017834123189888, −6.64877988680094143604691994840, −5.45607106090744336055785712714, −3.25177525130509162691033083574, −2.38843988022144906232913540507, −0.52024909961347291266822264363,
0.52024909961347291266822264363, 2.38843988022144906232913540507, 3.25177525130509162691033083574, 5.45607106090744336055785712714, 6.64877988680094143604691994840, 7.81194221508058017834123189888, 8.560024094341833702830684423424, 9.572940781948926011327783882000, 10.06873506978186275250001834568, 10.91522360646167686803765383689
