L(s) = 1 | + 3-s + 2.96·5-s − 3.61·7-s + 9-s − 1.51·11-s + 0.595·13-s + 2.96·15-s + 0.748·17-s − 3.13·19-s − 3.61·21-s + 23-s + 3.81·25-s + 27-s − 29-s + 1.50·31-s − 1.51·33-s − 10.7·35-s + 3.65·37-s + 0.595·39-s − 0.901·41-s − 7.23·43-s + 2.96·45-s + 6.53·47-s + 6.07·49-s + 0.748·51-s + 11.1·53-s − 4.50·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.32·5-s − 1.36·7-s + 0.333·9-s − 0.457·11-s + 0.165·13-s + 0.766·15-s + 0.181·17-s − 0.719·19-s − 0.788·21-s + 0.208·23-s + 0.763·25-s + 0.192·27-s − 0.185·29-s + 0.270·31-s − 0.264·33-s − 1.81·35-s + 0.601·37-s + 0.0954·39-s − 0.140·41-s − 1.10·43-s + 0.442·45-s + 0.953·47-s + 0.867·49-s + 0.104·51-s + 1.52·53-s − 0.607·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.699369504\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.699369504\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 - 2.96T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 - 0.595T + 13T^{2} \) |
| 17 | \( 1 - 0.748T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 0.901T + 41T^{2} \) |
| 43 | \( 1 + 7.23T + 43T^{2} \) |
| 47 | \( 1 - 6.53T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 4.00T + 61T^{2} \) |
| 67 | \( 1 - 7.19T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 - 8.91T + 73T^{2} \) |
| 79 | \( 1 - 8.63T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 7.05T + 97T^{2} \) |
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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
−7.940729565740858975984604709144, −6.80647526351939962056211456274, −6.65656729723729512419737906733, −5.73693678128754861870327811149, −5.27102979838816971135550929935, −4.11847598261819518822248610575, −3.39824165135558593801879354240, −2.54609175064899504735305159912, −2.07149862211835433337128095158, −0.77111572114126963912441107576,
0.77111572114126963912441107576, 2.07149862211835433337128095158, 2.54609175064899504735305159912, 3.39824165135558593801879354240, 4.11847598261819518822248610575, 5.27102979838816971135550929935, 5.73693678128754861870327811149, 6.65656729723729512419737906733, 6.80647526351939962056211456274, 7.940729565740858975984604709144