L(s) = 1 | − 0.481·3-s − 5-s − 7-s − 2.76·9-s − 11-s − 4.48·13-s + 0.481·15-s − 6.48·17-s − 3.73·19-s + 0.481·21-s − 5.11·23-s + 25-s + 2.77·27-s − 6.31·29-s − 0.899·31-s + 0.481·33-s + 35-s − 8.08·37-s + 2.15·39-s − 8.63·41-s + 7.50·43-s + 2.76·45-s + 10.4·47-s + 49-s + 3.11·51-s + 12.0·53-s + 55-s + ⋯ |
L(s) = 1 | − 0.277·3-s − 0.447·5-s − 0.377·7-s − 0.922·9-s − 0.301·11-s − 1.24·13-s + 0.124·15-s − 1.57·17-s − 0.857·19-s + 0.105·21-s − 1.06·23-s + 0.200·25-s + 0.534·27-s − 1.17·29-s − 0.161·31-s + 0.0837·33-s + 0.169·35-s − 1.32·37-s + 0.345·39-s − 1.34·41-s + 1.14·43-s + 0.412·45-s + 1.52·47-s + 0.142·49-s + 0.436·51-s + 1.65·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2177139352\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2177139352\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 0.481T + 3T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 5.11T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 + 0.899T + 31T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 + 8.63T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 + 0.712T + 61T^{2} \) |
| 67 | \( 1 + 0.932T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.743T + 73T^{2} \) |
| 79 | \( 1 - 1.96T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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
−8.118500396243414548469612104415, −7.19469274987421501940208295067, −6.78928593626077005773969853575, −5.83554351992354296269469398360, −5.29524248345623484175119374473, −4.37610108657723774420322203893, −3.75396981935793980323569855506, −2.61067365012167827557082318566, −2.10047095147797682612854569529, −0.22620074508143278786923845825,
0.22620074508143278786923845825, 2.10047095147797682612854569529, 2.61067365012167827557082318566, 3.75396981935793980323569855506, 4.37610108657723774420322203893, 5.29524248345623484175119374473, 5.83554351992354296269469398360, 6.78928593626077005773969853575, 7.19469274987421501940208295067, 8.118500396243414548469612104415