L(s) = 1 | + 2-s + 2.43·3-s + 4-s + 2.88·5-s + 2.43·6-s + 0.287·7-s + 8-s + 2.94·9-s + 2.88·10-s + 5.71·11-s + 2.43·12-s − 6.12·13-s + 0.287·14-s + 7.02·15-s + 16-s + 5.27·17-s + 2.94·18-s + 1.13·19-s + 2.88·20-s + 0.700·21-s + 5.71·22-s + 3.27·23-s + 2.43·24-s + 3.30·25-s − 6.12·26-s − 0.122·27-s + 0.287·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.40·3-s + 0.5·4-s + 1.28·5-s + 0.995·6-s + 0.108·7-s + 0.353·8-s + 0.983·9-s + 0.911·10-s + 1.72·11-s + 0.704·12-s − 1.69·13-s + 0.0767·14-s + 1.81·15-s + 0.250·16-s + 1.27·17-s + 0.695·18-s + 0.260·19-s + 0.644·20-s + 0.152·21-s + 1.21·22-s + 0.682·23-s + 0.497·24-s + 0.660·25-s − 1.20·26-s − 0.0236·27-s + 0.0542·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.777081524\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.777081524\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 - 0.287T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 - 1.13T + 19T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 0.910T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 + 5.48T + 67T^{2} \) |
| 71 | \( 1 + 6.21T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 1.05T + 83T^{2} \) |
| 89 | \( 1 + 0.512T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.525227670337103638274830083872, −7.59313161821298093080696371946, −7.02672698310075445823588067576, −6.24117358194903326913212913041, −5.35283926296658437240129245690, −4.70227018648306005584458387357, −3.51977850080257298208827300160, −3.12971434562439036442090082646, −1.99332803755348228446941199629, −1.58842122894774098818267682362,
1.58842122894774098818267682362, 1.99332803755348228446941199629, 3.12971434562439036442090082646, 3.51977850080257298208827300160, 4.70227018648306005584458387357, 5.35283926296658437240129245690, 6.24117358194903326913212913041, 7.02672698310075445823588067576, 7.59313161821298093080696371946, 8.525227670337103638274830083872