| L(s) = 1 | − 4.03·2-s + 8.31·4-s − 6.49·7-s − 1.27·8-s − 11·11-s + 30.5·13-s + 26.2·14-s − 61.3·16-s + 43.1·17-s − 6.46·19-s + 44.4·22-s − 108.·23-s − 123.·26-s − 54.0·28-s + 274.·29-s − 68.5·31-s + 258.·32-s − 174.·34-s − 402.·37-s + 26.0·38-s + 268.·41-s − 30.5·43-s − 91.4·44-s + 436.·46-s − 31.5·47-s − 300.·49-s + 253.·52-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.03·4-s − 0.350·7-s − 0.0564·8-s − 0.301·11-s + 0.651·13-s + 0.501·14-s − 0.958·16-s + 0.615·17-s − 0.0780·19-s + 0.430·22-s − 0.980·23-s − 0.930·26-s − 0.364·28-s + 1.75·29-s − 0.397·31-s + 1.42·32-s − 0.878·34-s − 1.78·37-s + 0.111·38-s + 1.02·41-s − 0.108·43-s − 0.313·44-s + 1.39·46-s − 0.0980·47-s − 0.876·49-s + 0.677·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.03T + 8T^{2} \) |
| 7 | \( 1 + 6.49T + 343T^{2} \) |
| 13 | \( 1 - 30.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.46T + 6.85e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 274.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 68.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 268.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 31.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 558.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 335.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 28.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 89.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 717.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 200.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 243.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 312.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 27.9T + 9.12e5T^{2} \) |
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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
−8.353269635687547896373284706757, −7.68689843590326600292642929956, −6.85983408587648447115154863048, −6.16322256798716487300231275938, −5.14508688907335366592253377243, −4.08011478932058509051533724222, −3.04222779260184957883982800014, −1.94673506044199708653465687081, −0.997909905006072522647684163779, 0,
0.997909905006072522647684163779, 1.94673506044199708653465687081, 3.04222779260184957883982800014, 4.08011478932058509051533724222, 5.14508688907335366592253377243, 6.16322256798716487300231275938, 6.85983408587648447115154863048, 7.68689843590326600292642929956, 8.353269635687547896373284706757
