L(s) = 1 | − 0.879·2-s − 1.22·4-s + 2·5-s − 2.87·7-s + 2.83·8-s − 1.75·10-s + 6.41·11-s + 0.694·13-s + 2.53·14-s − 0.0418·16-s − 1.06·17-s + 0.532·19-s − 2.45·20-s − 5.63·22-s − 7.00·23-s − 25-s − 0.610·26-s + 3.53·28-s + 3.38·29-s + 2.04·31-s − 5.63·32-s + 0.935·34-s − 5.75·35-s − 5.22·37-s − 0.467·38-s + 5.67·40-s + 11.4·41-s + ⋯ |
L(s) = 1 | − 0.621·2-s − 0.613·4-s + 0.894·5-s − 1.08·7-s + 1.00·8-s − 0.556·10-s + 1.93·11-s + 0.192·13-s + 0.676·14-s − 0.0104·16-s − 0.258·17-s + 0.122·19-s − 0.548·20-s − 1.20·22-s − 1.46·23-s − 0.200·25-s − 0.119·26-s + 0.667·28-s + 0.629·29-s + 0.366·31-s − 0.996·32-s + 0.160·34-s − 0.973·35-s − 0.859·37-s − 0.0759·38-s + 0.897·40-s + 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.145606083\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145606083\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 13 | \( 1 - 0.694T + 13T^{2} \) |
| 17 | \( 1 + 1.06T + 17T^{2} \) |
| 19 | \( 1 - 0.532T + 19T^{2} \) |
| 23 | \( 1 + 7.00T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 - 0.980T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 1.06T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 + 1.44T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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
−9.661841154828972345667485459521, −9.030597996836200200323031832632, −8.287837943190815824339361325945, −7.09620058782169407992918479458, −6.32909190915607288323359547639, −5.69858057716696411017188633829, −4.29592944660928590180879382064, −3.67142191129126448783723454526, −2.11940708019387281291893927569, −0.894994230460592668537197096810,
0.894994230460592668537197096810, 2.11940708019387281291893927569, 3.67142191129126448783723454526, 4.29592944660928590180879382064, 5.69858057716696411017188633829, 6.32909190915607288323359547639, 7.09620058782169407992918479458, 8.287837943190815824339361325945, 9.030597996836200200323031832632, 9.661841154828972345667485459521