L(s) = 1 | + 0.928·2-s − 3.32·3-s − 1.13·4-s − 5-s − 3.08·6-s − 7-s − 2.91·8-s + 8.05·9-s − 0.928·10-s + 4.79·11-s + 3.78·12-s − 5.29·13-s − 0.928·14-s + 3.32·15-s − 0.430·16-s − 3.68·17-s + 7.48·18-s − 0.548·19-s + 1.13·20-s + 3.32·21-s + 4.45·22-s − 2.08·23-s + 9.68·24-s + 25-s − 4.91·26-s − 16.8·27-s + 1.13·28-s + ⋯ |
L(s) = 1 | + 0.656·2-s − 1.91·3-s − 0.568·4-s − 0.447·5-s − 1.26·6-s − 0.377·7-s − 1.03·8-s + 2.68·9-s − 0.293·10-s + 1.44·11-s + 1.09·12-s − 1.46·13-s − 0.248·14-s + 0.858·15-s − 0.107·16-s − 0.894·17-s + 1.76·18-s − 0.125·19-s + 0.254·20-s + 0.725·21-s + 0.949·22-s − 0.434·23-s + 1.97·24-s + 0.200·25-s − 0.963·26-s − 3.23·27-s + 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 0.928T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 3.68T + 17T^{2} \) |
| 19 | \( 1 + 0.548T + 19T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 - 8.56T + 59T^{2} \) |
| 61 | \( 1 + 1.69T + 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 5.56T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.93649927858038258884291086561, −6.77298351133896205845804116863, −5.96505737272089562082338886055, −5.34961178390354161080654165007, −4.68781846611916685759552924003, −4.18248675764426709349412931907, −3.61186211132733644775679998226, −2.15134832137427753596153633567, −0.814742909003710656608148059481, 0,
0.814742909003710656608148059481, 2.15134832137427753596153633567, 3.61186211132733644775679998226, 4.18248675764426709349412931907, 4.68781846611916685759552924003, 5.34961178390354161080654165007, 5.96505737272089562082338886055, 6.77298351133896205845804116863, 6.93649927858038258884291086561