L(s) = 1 | − 2.26·2-s + 0.846·3-s + 3.13·4-s + 5-s − 1.91·6-s − 3.17·7-s − 2.56·8-s − 2.28·9-s − 2.26·10-s − 11-s + 2.65·12-s + 0.106·13-s + 7.19·14-s + 0.846·15-s − 0.460·16-s − 6.41·17-s + 5.17·18-s − 4.21·19-s + 3.13·20-s − 2.68·21-s + 2.26·22-s − 5.49·23-s − 2.16·24-s + 25-s − 0.241·26-s − 4.47·27-s − 9.93·28-s + ⋯ |
L(s) = 1 | − 1.60·2-s + 0.488·3-s + 1.56·4-s + 0.447·5-s − 0.783·6-s − 1.19·7-s − 0.905·8-s − 0.760·9-s − 0.716·10-s − 0.301·11-s + 0.765·12-s + 0.0295·13-s + 1.92·14-s + 0.218·15-s − 0.115·16-s − 1.55·17-s + 1.21·18-s − 0.966·19-s + 0.700·20-s − 0.586·21-s + 0.482·22-s − 1.14·23-s − 0.442·24-s + 0.200·25-s − 0.0473·26-s − 0.860·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3944877095\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3944877095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 0.846T + 3T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 13 | \( 1 - 0.106T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 0.824T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 - 5.77T + 61T^{2} \) |
| 67 | \( 1 + 9.02T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−8.496951267330266031896955069225, −8.144398272430530340547934287847, −7.04296877431436042345693669563, −6.49165391910068717019474614355, −5.93541747572840853052416992848, −4.63152336820201046461841046944, −3.50961818815289674795800529615, −2.47331476138439509212665787022, −2.05555899029502812271687483930, −0.41385974128026259819175343702,
0.41385974128026259819175343702, 2.05555899029502812271687483930, 2.47331476138439509212665787022, 3.50961818815289674795800529615, 4.63152336820201046461841046944, 5.93541747572840853052416992848, 6.49165391910068717019474614355, 7.04296877431436042345693669563, 8.144398272430530340547934287847, 8.496951267330266031896955069225