L(s) = 1 | + 2-s + 3-s + 4-s + 4.22·5-s + 6-s − 0.979·7-s + 8-s + 9-s + 4.22·10-s + 1.72·11-s + 12-s + 13-s − 0.979·14-s + 4.22·15-s + 16-s − 2.27·17-s + 18-s − 1.97·19-s + 4.22·20-s − 0.979·21-s + 1.72·22-s + 1.31·23-s + 24-s + 12.8·25-s + 26-s + 27-s − 0.979·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.88·5-s + 0.408·6-s − 0.370·7-s + 0.353·8-s + 0.333·9-s + 1.33·10-s + 0.519·11-s + 0.288·12-s + 0.277·13-s − 0.261·14-s + 1.09·15-s + 0.250·16-s − 0.550·17-s + 0.235·18-s − 0.454·19-s + 0.944·20-s − 0.213·21-s + 0.367·22-s + 0.273·23-s + 0.204·24-s + 2.56·25-s + 0.196·26-s + 0.192·27-s − 0.185·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.446936809\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.446936809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 + 0.979T + 7T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 - 9.39T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 5.91T + 73T^{2} \) |
| 79 | \( 1 + 8.55T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.80642319321901049220640581783, −6.73609056640165442860530348004, −6.30170128011914261096004430871, −6.03548477203638459926711534565, −4.81475876771048455598505409275, −4.55933024488524010087663264420, −3.24864123765960477878883715551, −2.76848882622406580841180289419, −1.93559769420035452853972063146, −1.21549216585654728629726439143,
1.21549216585654728629726439143, 1.93559769420035452853972063146, 2.76848882622406580841180289419, 3.24864123765960477878883715551, 4.55933024488524010087663264420, 4.81475876771048455598505409275, 6.03548477203638459926711534565, 6.30170128011914261096004430871, 6.73609056640165442860530348004, 7.80642319321901049220640581783