| L(s) = 1 | + 5.44·2-s + 21.6·4-s + 7·7-s + 74.3·8-s + 60.4·11-s + 64.0·13-s + 38.1·14-s + 231.·16-s − 37.7·17-s − 134.·19-s + 329.·22-s − 52.6·23-s + 348.·26-s + 151.·28-s + 165.·29-s − 71.9·31-s + 667.·32-s − 205.·34-s + 48.7·37-s − 734.·38-s − 10.5·41-s − 425.·43-s + 1.31e3·44-s − 286.·46-s − 249.·47-s + 49·49-s + 1.38e3·52-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.70·4-s + 0.377·7-s + 3.28·8-s + 1.65·11-s + 1.36·13-s + 0.727·14-s + 3.62·16-s − 0.538·17-s − 1.62·19-s + 3.19·22-s − 0.477·23-s + 2.63·26-s + 1.02·28-s + 1.05·29-s − 0.416·31-s + 3.68·32-s − 1.03·34-s + 0.216·37-s − 3.13·38-s − 0.0402·41-s − 1.51·43-s + 4.48·44-s − 0.918·46-s − 0.773·47-s + 0.142·49-s + 3.70·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.15046285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.15046285\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 5.44T + 8T^{2} \) |
| 11 | \( 1 - 60.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 52.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 71.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 48.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 425.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 544.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 567.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 614.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 201.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 525.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 137.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 234.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 10.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 20.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−8.856629402957580063645162903067, −8.167243995603459509441097506413, −6.82376220858614218452729041848, −6.48428252222754010003045532916, −5.81304940604708909520005336926, −4.66066281280979298168403826010, −4.07788820175730158385843391977, −3.44230435222000986648953939844, −2.14945308910623565157214415557, −1.34142716982583597837147846290,
1.34142716982583597837147846290, 2.14945308910623565157214415557, 3.44230435222000986648953939844, 4.07788820175730158385843391977, 4.66066281280979298168403826010, 5.81304940604708909520005336926, 6.48428252222754010003045532916, 6.82376220858614218452729041848, 8.167243995603459509441097506413, 8.856629402957580063645162903067
