| L(s) = 1 | + 2-s − 4-s + 5-s + 2·7-s − 3·8-s + 10-s + 4·11-s + 2·13-s + 2·14-s − 16-s − 6·17-s − 20-s + 4·22-s − 6·23-s + 25-s + 2·26-s − 2·28-s + 4·31-s + 5·32-s − 6·34-s + 2·35-s − 6·37-s − 3·40-s − 8·41-s + 6·43-s − 4·44-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s − 1.06·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.223·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 0.338·35-s − 0.986·37-s − 0.474·40-s − 1.24·41-s + 0.914·43-s − 0.603·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.04521848681010, −15.54351037775623, −14.92367303178029, −14.49857130154241, −13.81628369011653, −13.66028613207178, −13.12611542323919, −12.20295336866384, −11.92496339750403, −11.39101728046086, −10.57640262916763, −10.08684261085153, −9.212265335106553, −8.798083170750284, −8.486259856531610, −7.524504228286132, −6.740339583943055, −6.076679980863475, −5.780056004851628, −4.710612858711630, −4.447032904131237, −3.800140528674437, −2.987467267463211, −1.993490000075137, −1.315238064771515, 0,
1.315238064771515, 1.993490000075137, 2.987467267463211, 3.800140528674437, 4.447032904131237, 4.710612858711630, 5.780056004851628, 6.076679980863475, 6.740339583943055, 7.524504228286132, 8.486259856531610, 8.798083170750284, 9.212265335106553, 10.08684261085153, 10.57640262916763, 11.39101728046086, 11.92496339750403, 12.20295336866384, 13.12611542323919, 13.66028613207178, 13.81628369011653, 14.49857130154241, 14.92367303178029, 15.54351037775623, 16.04521848681010
