| L(s) = 1 | + 0.813·2-s − 7.33·4-s − 5·5-s − 12.4·8-s − 4.06·10-s − 18.9·11-s − 50.9·13-s + 48.5·16-s + 111.·17-s − 24.9·19-s + 36.6·20-s − 15.4·22-s + 104.·23-s + 25·25-s − 41.4·26-s − 150.·29-s − 24.4·31-s + 139.·32-s + 90.4·34-s + 90.6·37-s − 20.2·38-s + 62.4·40-s + 371.·41-s + 278.·43-s + 138.·44-s + 85.1·46-s − 520.·47-s + ⋯ |
| L(s) = 1 | + 0.287·2-s − 0.917·4-s − 0.447·5-s − 0.551·8-s − 0.128·10-s − 0.518·11-s − 1.08·13-s + 0.758·16-s + 1.58·17-s − 0.301·19-s + 0.410·20-s − 0.149·22-s + 0.948·23-s + 0.200·25-s − 0.312·26-s − 0.960·29-s − 0.141·31-s + 0.769·32-s + 0.456·34-s + 0.402·37-s − 0.0865·38-s + 0.246·40-s + 1.41·41-s + 0.986·43-s + 0.476·44-s + 0.272·46-s − 1.61·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.813T + 8T^{2} \) |
| 11 | \( 1 + 18.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 24.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 90.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 520.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 334.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 239.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 235.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 253.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 467.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 189.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 17.5T + 9.12e5T^{2} \) |
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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
−8.067604537919356682509632639726, −7.80245734101184190618925732001, −6.81633552605003431841078507106, −5.59485949939583302419678719169, −5.16756899412861581938169600056, −4.27746230276186238840621950709, −3.44734775091491278942599215992, −2.57477272474898286740624434814, −1.01071036561209563130923788536, 0,
1.01071036561209563130923788536, 2.57477272474898286740624434814, 3.44734775091491278942599215992, 4.27746230276186238840621950709, 5.16756899412861581938169600056, 5.59485949939583302419678719169, 6.81633552605003431841078507106, 7.80245734101184190618925732001, 8.067604537919356682509632639726
