L(s) = 1 | + 2-s + 0.790·3-s + 4-s − 3.26·5-s + 0.790·6-s − 4.99·7-s + 8-s − 2.37·9-s − 3.26·10-s − 0.714·11-s + 0.790·12-s − 5.95·13-s − 4.99·14-s − 2.57·15-s + 16-s − 5.93·17-s − 2.37·18-s − 19-s − 3.26·20-s − 3.95·21-s − 0.714·22-s − 6.38·23-s + 0.790·24-s + 5.64·25-s − 5.95·26-s − 4.24·27-s − 4.99·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.456·3-s + 0.5·4-s − 1.45·5-s + 0.322·6-s − 1.88·7-s + 0.353·8-s − 0.791·9-s − 1.03·10-s − 0.215·11-s + 0.228·12-s − 1.65·13-s − 1.33·14-s − 0.665·15-s + 0.250·16-s − 1.43·17-s − 0.559·18-s − 0.229·19-s − 0.729·20-s − 0.862·21-s − 0.152·22-s − 1.33·23-s + 0.161·24-s + 1.12·25-s − 1.16·26-s − 0.817·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1242257828\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1242257828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 0.790T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 + 4.99T + 7T^{2} \) |
| 11 | \( 1 + 0.714T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 - 0.168T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 6.98T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 8.63T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 8.50T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 4.91T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 0.0284T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 + 8.63T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.60759627287445096299467839999, −7.23586281836201308886320696506, −6.34952790547551173128577635629, −5.96633860509237405504497363925, −4.68592759160298009015531870482, −4.28476485368149795315280738232, −3.44031266152038266573799199968, −2.84475902095557058286942625174, −2.35172321043275337078944246138, −0.13652670460859259392615017373,
0.13652670460859259392615017373, 2.35172321043275337078944246138, 2.84475902095557058286942625174, 3.44031266152038266573799199968, 4.28476485368149795315280738232, 4.68592759160298009015531870482, 5.96633860509237405504497363925, 6.34952790547551173128577635629, 7.23586281836201308886320696506, 7.60759627287445096299467839999