L(s) = 1 | − 0.475·2-s − 1.77·4-s − 5-s + 2.49·7-s + 1.79·8-s + 0.475·10-s + 3.00·11-s + 13-s − 1.18·14-s + 2.69·16-s + 1.86·17-s − 6.61·19-s + 1.77·20-s − 1.42·22-s − 4.00·23-s + 25-s − 0.475·26-s − 4.42·28-s − 7.54·29-s − 1.10·31-s − 4.86·32-s − 0.885·34-s − 2.49·35-s − 10.2·37-s + 3.14·38-s − 1.79·40-s + 3.93·41-s + ⋯ |
L(s) = 1 | − 0.336·2-s − 0.887·4-s − 0.447·5-s + 0.943·7-s + 0.634·8-s + 0.150·10-s + 0.906·11-s + 0.277·13-s − 0.316·14-s + 0.673·16-s + 0.451·17-s − 1.51·19-s + 0.396·20-s − 0.304·22-s − 0.834·23-s + 0.200·25-s − 0.0932·26-s − 0.836·28-s − 1.40·29-s − 0.198·31-s − 0.860·32-s − 0.151·34-s − 0.421·35-s − 1.67·37-s + 0.510·38-s − 0.283·40-s + 0.614·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.475T + 2T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 4.00T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 - 0.132T + 47T^{2} \) |
| 53 | \( 1 - 6.91T + 53T^{2} \) |
| 59 | \( 1 + 0.804T + 59T^{2} \) |
| 61 | \( 1 + 4.78T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
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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
−7.88396171755958716160659104071, −7.42987008386717887322753318992, −6.40544208148873279210763097845, −5.58817177019965317663597066786, −4.80906664062591250378883457126, −4.01381779192349561126403451399, −3.69188578706498926453465355316, −2.09563829097388886651233020660, −1.26701543585755622062549107725, 0,
1.26701543585755622062549107725, 2.09563829097388886651233020660, 3.69188578706498926453465355316, 4.01381779192349561126403451399, 4.80906664062591250378883457126, 5.58817177019965317663597066786, 6.40544208148873279210763097845, 7.42987008386717887322753318992, 7.88396171755958716160659104071