| L(s) = 1 | + 0.429·2-s − 3-s − 1.81·4-s − 3.93·5-s − 0.429·6-s + 0.264·7-s − 1.63·8-s + 9-s − 1.68·10-s + 3.34·11-s + 1.81·12-s + 0.652·13-s + 0.113·14-s + 3.93·15-s + 2.92·16-s + 2.18·17-s + 0.429·18-s + 4.72·19-s + 7.13·20-s − 0.264·21-s + 1.43·22-s − 23-s + 1.63·24-s + 10.4·25-s + 0.280·26-s − 27-s − 0.480·28-s + ⋯ |
| L(s) = 1 | + 0.303·2-s − 0.577·3-s − 0.907·4-s − 1.75·5-s − 0.175·6-s + 0.0999·7-s − 0.579·8-s + 0.333·9-s − 0.534·10-s + 1.00·11-s + 0.524·12-s + 0.181·13-s + 0.0303·14-s + 1.01·15-s + 0.731·16-s + 0.530·17-s + 0.101·18-s + 1.08·19-s + 1.59·20-s − 0.0577·21-s + 0.306·22-s − 0.208·23-s + 0.334·24-s + 2.09·25-s + 0.0550·26-s − 0.192·27-s − 0.0907·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.429T + 2T^{2} \) |
| 5 | \( 1 + 3.93T + 5T^{2} \) |
| 7 | \( 1 - 0.264T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 0.652T + 13T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 0.521T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 7.56T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 5.97T + 97T^{2} \) |
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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
−8.698435178246431917831753535127, −7.974054742378939425099836257160, −7.27360779107979358504883077390, −6.39539186539259512979167359061, −5.26885942575556227286251786614, −4.70770758216030429154701285090, −3.62567312716857576773902694170, −3.52602327176977247970811710523, −1.19223370538469519432123302567, 0,
1.19223370538469519432123302567, 3.52602327176977247970811710523, 3.62567312716857576773902694170, 4.70770758216030429154701285090, 5.26885942575556227286251786614, 6.39539186539259512979167359061, 7.27360779107979358504883077390, 7.974054742378939425099836257160, 8.698435178246431917831753535127
