L(s) = 1 | + 2-s − 2.42·3-s + 4-s − 1.63·5-s − 2.42·6-s + 3.93·7-s + 8-s + 2.89·9-s − 1.63·10-s − 6.50·11-s − 2.42·12-s + 2.44·13-s + 3.93·14-s + 3.96·15-s + 16-s − 2.35·17-s + 2.89·18-s + 6.25·19-s − 1.63·20-s − 9.55·21-s − 6.50·22-s − 9.09·23-s − 2.42·24-s − 2.33·25-s + 2.44·26-s + 0.243·27-s + 3.93·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.40·3-s + 0.5·4-s − 0.729·5-s − 0.991·6-s + 1.48·7-s + 0.353·8-s + 0.966·9-s − 0.515·10-s − 1.96·11-s − 0.701·12-s + 0.677·13-s + 1.05·14-s + 1.02·15-s + 0.250·16-s − 0.570·17-s + 0.683·18-s + 1.43·19-s − 0.364·20-s − 2.08·21-s − 1.38·22-s − 1.89·23-s − 0.495·24-s − 0.467·25-s + 0.479·26-s + 0.0467·27-s + 0.743·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4001 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 - 5.51T + 41T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 + 8.11T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 2.22T + 67T^{2} \) |
| 71 | \( 1 + 2.67T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 7.38T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−7.64818672232699459408828131907, −6.54611802394877174246789910089, −5.90732702050331011071669850968, −5.30205825175249098478736197206, −4.73954065933813421126951033707, −4.35029286115591466477008323083, −3.21784328477181614867695619827, −2.23913912688963828415332479663, −1.16337108231943880519237913487, 0,
1.16337108231943880519237913487, 2.23913912688963828415332479663, 3.21784328477181614867695619827, 4.35029286115591466477008323083, 4.73954065933813421126951033707, 5.30205825175249098478736197206, 5.90732702050331011071669850968, 6.54611802394877174246789910089, 7.64818672232699459408828131907