L(s) = 1 | + 0.551·2-s − 1.69·4-s + 5-s + 1.42·7-s − 2.03·8-s + 0.551·10-s − 2.78·11-s − 4.03·13-s + 0.789·14-s + 2.26·16-s + 2.89·17-s + 3.03·19-s − 1.69·20-s − 1.53·22-s − 0.295·23-s + 25-s − 2.22·26-s − 2.42·28-s + 5.45·29-s − 9.06·31-s + 5.32·32-s + 1.59·34-s + 1.42·35-s + 7.41·37-s + 1.67·38-s − 2.03·40-s + 0.989·41-s + ⋯ |
L(s) = 1 | + 0.390·2-s − 0.847·4-s + 0.447·5-s + 0.540·7-s − 0.721·8-s + 0.174·10-s − 0.838·11-s − 1.11·13-s + 0.210·14-s + 0.566·16-s + 0.702·17-s + 0.696·19-s − 0.379·20-s − 0.327·22-s − 0.0616·23-s + 0.200·25-s − 0.437·26-s − 0.458·28-s + 1.01·29-s − 1.62·31-s + 0.942·32-s + 0.274·34-s + 0.241·35-s + 1.21·37-s + 0.271·38-s − 0.322·40-s + 0.154·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 - 0.551T + 2T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 + 0.295T + 23T^{2} \) |
| 29 | \( 1 - 5.45T + 29T^{2} \) |
| 31 | \( 1 + 9.06T + 31T^{2} \) |
| 37 | \( 1 - 7.41T + 37T^{2} \) |
| 41 | \( 1 - 0.989T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 8.42T + 59T^{2} \) |
| 61 | \( 1 + 0.169T + 61T^{2} \) |
| 67 | \( 1 + 1.78T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 + 8.61T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
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
−7.909692183901219748574800387157, −7.63045110235941789337701401628, −6.45678266204573874167895264741, −5.57351014379428989699486282998, −5.04468602780091973833114149391, −4.54528958754388020350103244935, −3.37674504593616552585568434540, −2.65894025277612503639466771564, −1.42550835299795628668105405635, 0,
1.42550835299795628668105405635, 2.65894025277612503639466771564, 3.37674504593616552585568434540, 4.54528958754388020350103244935, 5.04468602780091973833114149391, 5.57351014379428989699486282998, 6.45678266204573874167895264741, 7.63045110235941789337701401628, 7.909692183901219748574800387157