| L(s) = 1 | + 3-s − 1.51·5-s − 4.32·7-s + 9-s + 1.39·11-s + 0.610·13-s − 1.51·15-s + 3.34·17-s − 3.06·19-s − 4.32·21-s − 23-s − 2.70·25-s + 27-s − 29-s + 6.12·31-s + 1.39·33-s + 6.55·35-s + 9.13·37-s + 0.610·39-s − 4.45·41-s + 9.15·43-s − 1.51·45-s + 0.760·47-s + 11.6·49-s + 3.34·51-s + 2.50·53-s − 2.12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.677·5-s − 1.63·7-s + 0.333·9-s + 0.422·11-s + 0.169·13-s − 0.391·15-s + 0.810·17-s − 0.704·19-s − 0.943·21-s − 0.208·23-s − 0.541·25-s + 0.192·27-s − 0.185·29-s + 1.09·31-s + 0.243·33-s + 1.10·35-s + 1.50·37-s + 0.0977·39-s − 0.695·41-s + 1.39·43-s − 0.225·45-s + 0.110·47-s + 1.67·49-s + 0.468·51-s + 0.343·53-s − 0.285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 0.610T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 31 | \( 1 - 6.12T + 31T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 - 9.15T + 43T^{2} \) |
| 47 | \( 1 - 0.760T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 + 0.00974T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 1.88T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.51775238168418995505295968227, −6.82701941990525928384708443276, −6.20896280541604563904526608244, −5.57863819900523891798527973789, −4.17883532481086797660721716832, −4.00948719971395893061608984674, −3.07357830206812076144304727332, −2.54355357792902980975192450724, −1.16650593626819708263765647891, 0,
1.16650593626819708263765647891, 2.54355357792902980975192450724, 3.07357830206812076144304727332, 4.00948719971395893061608984674, 4.17883532481086797660721716832, 5.57863819900523891798527973789, 6.20896280541604563904526608244, 6.82701941990525928384708443276, 7.51775238168418995505295968227
