| L(s) = 1 | + 2-s − 2.60·3-s + 4-s − 0.616·5-s − 2.60·6-s − 3.04·7-s + 8-s + 3.79·9-s − 0.616·10-s − 0.708·11-s − 2.60·12-s + 4.52·13-s − 3.04·14-s + 1.60·15-s + 16-s − 7.53·17-s + 3.79·18-s + 1.15·19-s − 0.616·20-s + 7.93·21-s − 0.708·22-s + 6.01·23-s − 2.60·24-s − 4.61·25-s + 4.52·26-s − 2.06·27-s − 3.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s − 0.275·5-s − 1.06·6-s − 1.15·7-s + 0.353·8-s + 1.26·9-s − 0.195·10-s − 0.213·11-s − 0.752·12-s + 1.25·13-s − 0.813·14-s + 0.415·15-s + 0.250·16-s − 1.82·17-s + 0.894·18-s + 0.264·19-s − 0.137·20-s + 1.73·21-s − 0.151·22-s + 1.25·23-s − 0.532·24-s − 0.923·25-s + 0.886·26-s − 0.398·27-s − 0.575·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 5 | \( 1 + 0.616T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 0.708T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 - 3.05T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 - 0.716T + 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 - 1.06T + 43T^{2} \) |
| 47 | \( 1 - 5.03T + 47T^{2} \) |
| 53 | \( 1 + 6.45T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 + 1.93T + 61T^{2} \) |
| 67 | \( 1 + 2.98T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 5.50T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 - 2.45T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.02603116251637525036571057010, −6.55282009475909868982910694807, −6.14120579619048013028105106315, −5.47926733303746687426060417824, −4.74023164717287388720659988171, −4.03633618338505222045745875195, −3.34216001545315296925023329953, −2.32933463438924950126679845701, −1.02901697468313988989131593572, 0,
1.02901697468313988989131593572, 2.32933463438924950126679845701, 3.34216001545315296925023329953, 4.03633618338505222045745875195, 4.74023164717287388720659988171, 5.47926733303746687426060417824, 6.14120579619048013028105106315, 6.55282009475909868982910694807, 7.02603116251637525036571057010
