| L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·11-s + 2·13-s − 6·19-s + 4·21-s − 27-s − 10·29-s + 8·31-s − 2·33-s − 4·37-s − 2·39-s − 10·41-s − 43-s + 9·49-s + 12·53-s + 6·57-s + 6·59-s + 10·61-s − 4·63-s + 12·67-s + 4·71-s + 8·73-s − 8·77-s − 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.37·19-s + 0.872·21-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.348·33-s − 0.657·37-s − 0.320·39-s − 1.56·41-s − 0.152·43-s + 9/7·49-s + 1.64·53-s + 0.794·57-s + 0.781·59-s + 1.28·61-s − 0.503·63-s + 1.46·67-s + 0.474·71-s + 0.936·73-s − 0.911·77-s − 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.19626155593900, −12.75450843868210, −12.45734091474059, −11.84643848029035, −11.37521525098599, −11.04537240356770, −10.33117481750624, −9.919880608844507, −9.773969594973642, −8.974804956826022, −8.567995217064686, −8.243482378130690, −7.256982501367134, −6.883009149009714, −6.584262646598581, −6.103090335893200, −5.585602939147343, −5.144193833843691, −4.255757576387336, −3.854858004504746, −3.544201023251594, −2.742305773669748, −2.141020958951430, −1.438896767086288, −0.6247728361318038, 0,
0.6247728361318038, 1.438896767086288, 2.141020958951430, 2.742305773669748, 3.544201023251594, 3.854858004504746, 4.255757576387336, 5.144193833843691, 5.585602939147343, 6.103090335893200, 6.584262646598581, 6.883009149009714, 7.256982501367134, 8.243482378130690, 8.567995217064686, 8.974804956826022, 9.773969594973642, 9.919880608844507, 10.33117481750624, 11.04537240356770, 11.37521525098599, 11.84643848029035, 12.45734091474059, 12.75450843868210, 13.19626155593900
