| L(s) = 1 | + 2·2-s + 2·4-s − 3·7-s − 2·13-s − 6·14-s − 4·16-s + 6·17-s + 5·19-s − 4·23-s − 4·26-s − 6·28-s − 6·29-s − 5·31-s − 8·32-s + 12·34-s + 11·37-s + 10·38-s − 4·41-s − 4·43-s − 8·46-s + 6·47-s + 2·49-s − 4·52-s + 8·53-s − 12·58-s − 2·59-s − 61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.13·7-s − 0.554·13-s − 1.60·14-s − 16-s + 1.45·17-s + 1.14·19-s − 0.834·23-s − 0.784·26-s − 1.13·28-s − 1.11·29-s − 0.898·31-s − 1.41·32-s + 2.05·34-s + 1.80·37-s + 1.62·38-s − 0.624·41-s − 0.609·43-s − 1.17·46-s + 0.875·47-s + 2/7·49-s − 0.554·52-s + 1.09·53-s − 1.57·58-s − 0.260·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.196949147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.196949147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−15.04357047648338, −14.64560430955452, −14.20368530178503, −13.59019781371622, −13.12684415336249, −12.73469407629422, −12.12919118970699, −11.79420123665651, −11.25361577856727, −10.32413040082269, −9.840775889345113, −9.424932582577595, −8.809554304302918, −7.734665411533695, −7.444151627776026, −6.727203617901746, −6.028685587980148, −5.606454141083555, −5.197288059155496, −4.274453472062379, −3.738804717933254, −3.179522997827753, −2.704906962348526, −1.749816573235863, −0.5330485209481303,
0.5330485209481303, 1.749816573235863, 2.704906962348526, 3.179522997827753, 3.738804717933254, 4.274453472062379, 5.197288059155496, 5.606454141083555, 6.028685587980148, 6.727203617901746, 7.444151627776026, 7.734665411533695, 8.809554304302918, 9.424932582577595, 9.840775889345113, 10.32413040082269, 11.25361577856727, 11.79420123665651, 12.12919118970699, 12.73469407629422, 13.12684415336249, 13.59019781371622, 14.20368530178503, 14.64560430955452, 15.04357047648338
