L(s) = 1 | − 2-s + 0.0970·3-s + 4-s + 5-s − 0.0970·6-s + 7-s − 8-s − 2.99·9-s − 10-s + 0.0970·12-s + 4.32·13-s − 14-s + 0.0970·15-s + 16-s − 2.12·17-s + 2.99·18-s − 5.51·19-s + 20-s + 0.0970·21-s + 0.922·23-s − 0.0970·24-s + 25-s − 4.32·26-s − 0.581·27-s + 28-s + 8.02·29-s − 0.0970·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.0560·3-s + 0.5·4-s + 0.447·5-s − 0.0396·6-s + 0.377·7-s − 0.353·8-s − 0.996·9-s − 0.316·10-s + 0.0280·12-s + 1.19·13-s − 0.267·14-s + 0.0250·15-s + 0.250·16-s − 0.514·17-s + 0.704·18-s − 1.26·19-s + 0.223·20-s + 0.0211·21-s + 0.192·23-s − 0.0198·24-s + 0.200·25-s − 0.848·26-s − 0.111·27-s + 0.188·28-s + 1.49·29-s − 0.0177·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.474309693\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.474309693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.0970T + 3T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 - 0.922T + 23T^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 + 0.865T + 47T^{2} \) |
| 53 | \( 1 + 0.729T + 53T^{2} \) |
| 59 | \( 1 + 0.178T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 2.63T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 0.0855T + 97T^{2} \) |
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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
−8.160075381376608599470349136039, −7.01649275359835722235543013150, −6.45516348183814286817245293121, −5.91876211995740533488658638841, −5.12789122208833622471543813452, −4.24259366254694932448373950986, −3.29509546433392112248891116559, −2.47959415152666095479275985882, −1.72692576764083500861062311709, −0.65964727515123509710315293016,
0.65964727515123509710315293016, 1.72692576764083500861062311709, 2.47959415152666095479275985882, 3.29509546433392112248891116559, 4.24259366254694932448373950986, 5.12789122208833622471543813452, 5.91876211995740533488658638841, 6.45516348183814286817245293121, 7.01649275359835722235543013150, 8.160075381376608599470349136039