| L(s) = 1 | − 3·3-s + 3.46·5-s + 24.2·7-s + 9·9-s − 48·11-s + 41.5·13-s − 10.3·15-s + 54·17-s + 4·19-s − 72.7·21-s + 173.·23-s − 113·25-s − 27·27-s + 162.·29-s − 58.8·31-s + 144·33-s + 84·35-s − 325.·37-s − 124.·39-s + 294·41-s − 188·43-s + 31.1·45-s − 505.·47-s + 245·49-s − 162·51-s + 744.·53-s − 166.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.309·5-s + 1.30·7-s + 0.333·9-s − 1.31·11-s + 0.886·13-s − 0.178·15-s + 0.770·17-s + 0.0482·19-s − 0.755·21-s + 1.57·23-s − 0.904·25-s − 0.192·27-s + 1.04·29-s − 0.341·31-s + 0.759·33-s + 0.405·35-s − 1.44·37-s − 0.512·39-s + 1.11·41-s − 0.666·43-s + 0.103·45-s − 1.56·47-s + 0.714·49-s − 0.444·51-s + 1.93·53-s − 0.407·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.089822593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089822593\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 3.46T + 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 + 48T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 294T + 6.89e4T^{2} \) |
| 43 | \( 1 + 188T + 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 252T + 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 628T + 3.00e5T^{2} \) |
| 71 | \( 1 + 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 720T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−10.23769701594474768655671283090, −9.006663934073312732513676519477, −8.116529215936313770241247597135, −7.46358609558734544415614344403, −6.28723739653473530039479463943, −5.26706013733793090386031089710, −4.87714819642146302850777356992, −3.41023000613132208474580890528, −2.00271603941077803915569238748, −0.878320722141562171371848981036,
0.878320722141562171371848981036, 2.00271603941077803915569238748, 3.41023000613132208474580890528, 4.87714819642146302850777356992, 5.26706013733793090386031089710, 6.28723739653473530039479463943, 7.46358609558734544415614344403, 8.116529215936313770241247597135, 9.006663934073312732513676519477, 10.23769701594474768655671283090
