| L(s) = 1 | + 0.258·2-s − 7.93·4-s − 14.5·7-s − 4.12·8-s − 49.2·11-s − 72.1·13-s − 3.75·14-s + 62.3·16-s − 118.·17-s + 123.·19-s − 12.7·22-s + 91.4·23-s − 18.6·26-s + 115.·28-s + 174.·29-s − 46.2·31-s + 49.1·32-s − 30.5·34-s − 154.·37-s + 31.9·38-s − 364.·41-s − 125.·43-s + 390.·44-s + 23.6·46-s + 221.·47-s − 132.·49-s + 572.·52-s + ⋯ |
| L(s) = 1 | + 0.0914·2-s − 0.991·4-s − 0.783·7-s − 0.182·8-s − 1.35·11-s − 1.53·13-s − 0.0716·14-s + 0.974·16-s − 1.68·17-s + 1.48·19-s − 0.123·22-s + 0.829·23-s − 0.140·26-s + 0.777·28-s + 1.11·29-s − 0.268·31-s + 0.271·32-s − 0.154·34-s − 0.688·37-s + 0.136·38-s − 1.38·41-s − 0.445·43-s + 1.33·44-s + 0.0758·46-s + 0.687·47-s − 0.385·49-s + 1.52·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6430716404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6430716404\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.258T + 8T^{2} \) |
| 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 + 49.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 13.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 76.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 335.T + 9.12e5T^{2} \) |
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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
−9.942352266062744948490247823560, −9.367073117954284820464194773596, −8.457948205417596557045396508922, −7.48513746218247372118296796378, −6.61654057060561390362463320771, −5.13965622973339855034720923065, −4.91206280431900223880113300253, −3.43242353981598250967734743838, −2.48000109229321101592908827501, −0.43724613678547454417322215859,
0.43724613678547454417322215859, 2.48000109229321101592908827501, 3.43242353981598250967734743838, 4.91206280431900223880113300253, 5.13965622973339855034720923065, 6.61654057060561390362463320771, 7.48513746218247372118296796378, 8.457948205417596557045396508922, 9.367073117954284820464194773596, 9.942352266062744948490247823560
