| L(s) = 1 | − 109.·3-s − 2.49e3·5-s − 2.87e3·7-s − 7.64e3·9-s − 3.65e4·11-s + 6.99e4·13-s + 2.74e5·15-s + 8.35e4·17-s − 6.40e5·19-s + 3.15e5·21-s − 2.09e6·23-s + 4.28e6·25-s + 2.99e6·27-s + 4.99e6·29-s + 5.61e6·31-s + 4.01e6·33-s + 7.17e6·35-s + 3.47e6·37-s − 7.67e6·39-s + 4.69e5·41-s − 3.50e6·43-s + 1.90e7·45-s − 1.55e6·47-s − 3.21e7·49-s − 9.16e6·51-s + 1.03e8·53-s + 9.13e7·55-s + ⋯ |
| L(s) = 1 | − 0.782·3-s − 1.78·5-s − 0.452·7-s − 0.388·9-s − 0.753·11-s + 0.678·13-s + 1.39·15-s + 0.242·17-s − 1.12·19-s + 0.353·21-s − 1.55·23-s + 2.19·25-s + 1.08·27-s + 1.31·29-s + 1.09·31-s + 0.589·33-s + 0.808·35-s + 0.305·37-s − 0.530·39-s + 0.0259·41-s − 0.156·43-s + 0.693·45-s − 0.0464·47-s − 0.795·49-s − 0.189·51-s + 1.80·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 8.35e4T \) |
| good | 3 | \( 1 + 109.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.49e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.99e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 6.40e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.09e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.61e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.69e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.50e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.55e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.03e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.79e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.79e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.26e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.03e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.43e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.95e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.49e8T + 7.60e17T^{2} \) |
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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
−10.19619338674877773618280025381, −8.455669408780945616622593001662, −8.123163587006298411019566286556, −6.83227206838789673529725969323, −5.95578383393949300886449671871, −4.68378183181592491494652694145, −3.81648204094233093013733358956, −2.70714328810356266344300394109, −0.73579777085718367641466921887, 0,
0.73579777085718367641466921887, 2.70714328810356266344300394109, 3.81648204094233093013733358956, 4.68378183181592491494652694145, 5.95578383393949300886449671871, 6.83227206838789673529725969323, 8.123163587006298411019566286556, 8.455669408780945616622593001662, 10.19619338674877773618280025381
