L(s) = 1 | − 2-s − 0.934·3-s + 4-s − 4.17·5-s + 0.934·6-s + 1.30·7-s − 8-s − 2.12·9-s + 4.17·10-s − 6.29·11-s − 0.934·12-s − 3.79·13-s − 1.30·14-s + 3.89·15-s + 16-s + 3.77·17-s + 2.12·18-s + 8.63·19-s − 4.17·20-s − 1.21·21-s + 6.29·22-s − 0.896·23-s + 0.934·24-s + 12.3·25-s + 3.79·26-s + 4.79·27-s + 1.30·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.539·3-s + 0.5·4-s − 1.86·5-s + 0.381·6-s + 0.493·7-s − 0.353·8-s − 0.709·9-s + 1.31·10-s − 1.89·11-s − 0.269·12-s − 1.05·13-s − 0.348·14-s + 1.00·15-s + 0.250·16-s + 0.916·17-s + 0.501·18-s + 1.98·19-s − 0.932·20-s − 0.266·21-s + 1.34·22-s − 0.186·23-s + 0.190·24-s + 2.47·25-s + 0.744·26-s + 0.921·27-s + 0.246·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 0.934T + 3T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 8.63T + 19T^{2} \) |
| 23 | \( 1 + 0.896T + 23T^{2} \) |
| 29 | \( 1 + 5.01T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 0.395T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 - 5.28T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 - 7.71T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 5.22T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 4.55T + 97T^{2} \) |
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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
−8.005282176378709724217000206624, −7.49211859915586737515021374545, −7.13800582715077088372596315439, −5.55560691820766118225680389450, −5.30300333357420222207774960562, −4.35473872468671227927173459068, −3.14958065413621810930113191923, −2.67612011449580629870559458716, −0.865145251846734321892975022309, 0,
0.865145251846734321892975022309, 2.67612011449580629870559458716, 3.14958065413621810930113191923, 4.35473872468671227927173459068, 5.30300333357420222207774960562, 5.55560691820766118225680389450, 7.13800582715077088372596315439, 7.49211859915586737515021374545, 8.005282176378709724217000206624