L(s) = 1 | − 1.98·2-s + 1.93·4-s + 0.0269·5-s + 7-s + 0.122·8-s − 0.0534·10-s − 4.88·13-s − 1.98·14-s − 4.11·16-s − 1.67·17-s + 1.37·19-s + 0.0521·20-s − 8.06·23-s − 4.99·25-s + 9.68·26-s + 1.93·28-s + 6.39·29-s + 4.01·31-s + 7.93·32-s + 3.33·34-s + 0.0269·35-s + 0.521·37-s − 2.72·38-s + 0.00329·40-s + 10.5·41-s + 3.73·43-s + 16.0·46-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 0.969·4-s + 0.0120·5-s + 0.377·7-s + 0.0432·8-s − 0.0168·10-s − 1.35·13-s − 0.530·14-s − 1.02·16-s − 0.407·17-s + 0.315·19-s + 0.0116·20-s − 1.68·23-s − 0.999·25-s + 1.89·26-s + 0.366·28-s + 1.18·29-s + 0.720·31-s + 1.40·32-s + 0.571·34-s + 0.00455·35-s + 0.0856·37-s − 0.442·38-s + 0.000521·40-s + 1.65·41-s + 0.570·43-s + 2.35·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 5 | \( 1 - 0.0269T + 5T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + 8.06T + 23T^{2} \) |
| 29 | \( 1 - 6.39T + 29T^{2} \) |
| 31 | \( 1 - 4.01T + 31T^{2} \) |
| 37 | \( 1 - 0.521T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 - 3.96T + 53T^{2} \) |
| 59 | \( 1 + 9.73T + 59T^{2} \) |
| 61 | \( 1 - 8.46T + 61T^{2} \) |
| 67 | \( 1 - 2.81T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 + 2.60T + 83T^{2} \) |
| 89 | \( 1 + 1.21T + 89T^{2} \) |
| 97 | \( 1 - 3.39T + 97T^{2} \) |
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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
−7.74393139119881470042606485788, −7.18544858521687424797020322240, −6.33918791419430626287698363453, −5.57006128654203523431771221232, −4.56244287811666354746431004779, −4.10582482168397100557023076288, −2.60610662173162115953789937561, −2.14861279764480641867310182757, −1.03555223875678682698222945060, 0,
1.03555223875678682698222945060, 2.14861279764480641867310182757, 2.60610662173162115953789937561, 4.10582482168397100557023076288, 4.56244287811666354746431004779, 5.57006128654203523431771221232, 6.33918791419430626287698363453, 7.18544858521687424797020322240, 7.74393139119881470042606485788