| L(s) = 1 | + 3·3-s − 17.9·5-s + 31.7·7-s + 9·9-s − 28.3·11-s + 22.3·13-s − 53.9·15-s + 32.5·17-s − 9.45·19-s + 95.3·21-s − 23·23-s + 198.·25-s + 27·27-s − 87.3·29-s − 97.7·31-s − 85.1·33-s − 571.·35-s + 412.·37-s + 66.9·39-s + 184.·41-s + 400.·43-s − 161.·45-s + 98.3·47-s + 666.·49-s + 97.5·51-s + 253.·53-s + 510.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.60·5-s + 1.71·7-s + 0.333·9-s − 0.778·11-s + 0.476·13-s − 0.928·15-s + 0.463·17-s − 0.114·19-s + 0.990·21-s − 0.208·23-s + 1.58·25-s + 0.192·27-s − 0.559·29-s − 0.566·31-s − 0.449·33-s − 2.75·35-s + 1.83·37-s + 0.274·39-s + 0.703·41-s + 1.42·43-s − 0.536·45-s + 0.305·47-s + 1.94·49-s + 0.267·51-s + 0.657·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.157046255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.157046255\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 + 17.9T + 125T^{2} \) |
| 7 | \( 1 - 31.7T + 343T^{2} \) |
| 11 | \( 1 + 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.45T + 6.85e3T^{2} \) |
| 29 | \( 1 + 87.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 97.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 412.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 400.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 98.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 253.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 492.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 28.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 705.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 185.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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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
−10.75471629425606304052864444271, −9.277644857161777240703628503042, −8.165443937671676471389931585338, −7.971542912830049580986841781754, −7.25127112802657235295347062677, −5.54235296126913952331854352100, −4.47075411677084912744371364041, −3.79597971478023908473975214050, −2.40404557802670881180393009788, −0.899328315291421341168277828585,
0.899328315291421341168277828585, 2.40404557802670881180393009788, 3.79597971478023908473975214050, 4.47075411677084912744371364041, 5.54235296126913952331854352100, 7.25127112802657235295347062677, 7.971542912830049580986841781754, 8.165443937671676471389931585338, 9.277644857161777240703628503042, 10.75471629425606304052864444271
