L(s) = 1 | − 2-s − 1.09·3-s + 4-s − 0.358·5-s + 1.09·6-s − 2.93·7-s − 8-s − 1.79·9-s + 0.358·10-s − 2.70·11-s − 1.09·12-s + 0.734·13-s + 2.93·14-s + 0.393·15-s + 16-s + 0.422·17-s + 1.79·18-s + 8.46·19-s − 0.358·20-s + 3.21·21-s + 2.70·22-s − 8.04·23-s + 1.09·24-s − 4.87·25-s − 0.734·26-s + 5.25·27-s − 2.93·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.632·3-s + 0.5·4-s − 0.160·5-s + 0.447·6-s − 1.10·7-s − 0.353·8-s − 0.599·9-s + 0.113·10-s − 0.814·11-s − 0.316·12-s + 0.203·13-s + 0.783·14-s + 0.101·15-s + 0.250·16-s + 0.102·17-s + 0.423·18-s + 1.94·19-s − 0.0801·20-s + 0.700·21-s + 0.576·22-s − 1.67·23-s + 0.223·24-s − 0.974·25-s − 0.144·26-s + 1.01·27-s − 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2958611754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2958611754\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 + 0.358T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 - 0.734T + 13T^{2} \) |
| 17 | \( 1 - 0.422T + 17T^{2} \) |
| 19 | \( 1 - 8.46T + 19T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 + 3.52T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 0.269T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 2.29T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 7.19T + 79T^{2} \) |
| 83 | \( 1 + 2.45T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 + 7.90T + 97T^{2} \) |
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
−8.360133825253820722929920520985, −7.74619889637818839991150636622, −7.08472487337152806567955169536, −6.05326993324415750079860596583, −5.82560087117676928973659928562, −4.89083345081967114247851975953, −3.51931032813186989225899233048, −3.03550154057286062685792888880, −1.80272101936683949071532318888, −0.34128000299908444164992810245,
0.34128000299908444164992810245, 1.80272101936683949071532318888, 3.03550154057286062685792888880, 3.51931032813186989225899233048, 4.89083345081967114247851975953, 5.82560087117676928973659928562, 6.05326993324415750079860596583, 7.08472487337152806567955169536, 7.74619889637818839991150636622, 8.360133825253820722929920520985