| L(s) = 1 | + 0.460·2-s − 1.78·4-s − 4.10·5-s + 7-s − 1.74·8-s − 1.88·10-s − 2.76·11-s − 4.05·13-s + 0.460·14-s + 2.77·16-s − 5.22·17-s − 0.109·19-s + 7.33·20-s − 1.27·22-s − 6.08·23-s + 11.8·25-s − 1.86·26-s − 1.78·28-s − 2.25·29-s − 1.18·31-s + 4.76·32-s − 2.40·34-s − 4.10·35-s − 8.95·37-s − 0.0502·38-s + 7.15·40-s + 41-s + ⋯ |
| L(s) = 1 | + 0.325·2-s − 0.894·4-s − 1.83·5-s + 0.377·7-s − 0.616·8-s − 0.597·10-s − 0.834·11-s − 1.12·13-s + 0.123·14-s + 0.693·16-s − 1.26·17-s − 0.0250·19-s + 1.63·20-s − 0.271·22-s − 1.26·23-s + 2.36·25-s − 0.366·26-s − 0.337·28-s − 0.418·29-s − 0.212·31-s + 0.842·32-s − 0.412·34-s − 0.693·35-s − 1.47·37-s − 0.00815·38-s + 1.13·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3256887311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3256887311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.460T + 2T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 0.109T + 19T^{2} \) |
| 23 | \( 1 + 6.08T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 0.852T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−8.599869366323706424671380417526, −8.188776103238907716915753859232, −7.50654029442692881792773116313, −6.77831230735893453924836265344, −5.41277194758865571443177076837, −4.80721592155265660581996561574, −4.14240518196456688220343431743, −3.49440172688911640791885968725, −2.32425975236011734928205809944, −0.32437105844160595261655921017,
0.32437105844160595261655921017, 2.32425975236011734928205809944, 3.49440172688911640791885968725, 4.14240518196456688220343431743, 4.80721592155265660581996561574, 5.41277194758865571443177076837, 6.77831230735893453924836265344, 7.50654029442692881792773116313, 8.188776103238907716915753859232, 8.599869366323706424671380417526
