L(s) = 1 | − 0.834·2-s − 3-s − 1.30·4-s + 5-s + 0.834·6-s + 4.45·7-s + 2.75·8-s + 9-s − 0.834·10-s + 1.69·11-s + 1.30·12-s − 13-s − 3.71·14-s − 15-s + 0.309·16-s − 1.06·17-s − 0.834·18-s + 7.64·19-s − 1.30·20-s − 4.45·21-s − 1.41·22-s + 4.61·23-s − 2.75·24-s + 25-s + 0.834·26-s − 27-s − 5.81·28-s + ⋯ |
L(s) = 1 | − 0.589·2-s − 0.577·3-s − 0.652·4-s + 0.447·5-s + 0.340·6-s + 1.68·7-s + 0.974·8-s + 0.333·9-s − 0.263·10-s + 0.510·11-s + 0.376·12-s − 0.277·13-s − 0.994·14-s − 0.258·15-s + 0.0773·16-s − 0.258·17-s − 0.196·18-s + 1.75·19-s − 0.291·20-s − 0.973·21-s − 0.300·22-s + 0.962·23-s − 0.562·24-s + 0.200·25-s + 0.163·26-s − 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.659858305\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.659858305\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 0.834T + 2T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 17 | \( 1 + 1.06T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 5.06T + 43T^{2} \) |
| 47 | \( 1 - 7.22T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 4.04T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.990899451376907301957912667679, −7.53751014739497323418152929440, −6.88369717523554988296915096412, −5.59964856861061681364118568322, −5.31563603339326263361983472570, −4.56305259010224337106445486092, −3.93492614680002642474282059205, −2.52361033043648769907978328567, −1.38625312540931811283499341649, −0.917691169429319807777340723440,
0.917691169429319807777340723440, 1.38625312540931811283499341649, 2.52361033043648769907978328567, 3.93492614680002642474282059205, 4.56305259010224337106445486092, 5.31563603339326263361983472570, 5.59964856861061681364118568322, 6.88369717523554988296915096412, 7.53751014739497323418152929440, 7.990899451376907301957912667679