L(s) = 1 | − 0.618·2-s − 1.61·4-s − 2.28·5-s + 2.23·8-s + 1.41·10-s + 11-s − 4.57·13-s + 1.85·16-s + 3.16·17-s + 4.03·19-s + 3.70·20-s − 0.618·22-s − 7.23·23-s + 0.236·25-s + 2.82·26-s + 1.23·29-s + 5.99·31-s − 5.61·32-s − 1.95·34-s − 10.7·37-s − 2.49·38-s − 5.11·40-s + 5.78·41-s + 5.94·43-s − 1.61·44-s + 4.47·46-s + 3.03·47-s + ⋯ |
L(s) = 1 | − 0.437·2-s − 0.809·4-s − 1.02·5-s + 0.790·8-s + 0.447·10-s + 0.301·11-s − 1.26·13-s + 0.463·16-s + 0.766·17-s + 0.925·19-s + 0.827·20-s − 0.131·22-s − 1.50·23-s + 0.0472·25-s + 0.554·26-s + 0.229·29-s + 1.07·31-s − 0.993·32-s − 0.335·34-s − 1.76·37-s − 0.404·38-s − 0.809·40-s + 0.903·41-s + 0.906·43-s − 0.243·44-s + 0.659·46-s + 0.442·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 - 5.47T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 - 8.61T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−8.015404085966748123152822807834, −7.61366586905601631214974495888, −6.93654511565299587938198312839, −5.69142955341112128936953969223, −5.04015321128888718628311227855, −4.13787546287448282545836021119, −3.67858528210579546848526599244, −2.47215739719307441325815145112, −1.06432943688538997592263563449, 0,
1.06432943688538997592263563449, 2.47215739719307441325815145112, 3.67858528210579546848526599244, 4.13787546287448282545836021119, 5.04015321128888718628311227855, 5.69142955341112128936953969223, 6.93654511565299587938198312839, 7.61366586905601631214974495888, 8.015404085966748123152822807834