| L(s) = 1 | − 5-s − 7-s − 3·9-s + 5·11-s − 13-s − 4·19-s + 3·23-s + 25-s + 5·29-s − 5·31-s + 35-s + 2·37-s + 8·41-s − 6·43-s + 3·45-s − 6·49-s + 10·53-s − 5·55-s + 7·59-s − 14·61-s + 3·63-s + 65-s − 5·67-s + 6·71-s − 5·73-s − 5·77-s − 14·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 1.50·11-s − 0.277·13-s − 0.917·19-s + 0.625·23-s + 1/5·25-s + 0.928·29-s − 0.898·31-s + 0.169·35-s + 0.328·37-s + 1.24·41-s − 0.914·43-s + 0.447·45-s − 6/7·49-s + 1.37·53-s − 0.674·55-s + 0.911·59-s − 1.79·61-s + 0.377·63-s + 0.124·65-s − 0.610·67-s + 0.712·71-s − 0.585·73-s − 0.569·77-s − 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72099804916880757403741733197, −6.91252972203290249423273095538, −6.37047528066041968073239420950, −5.70196145539737324099653241529, −4.71711643897163330818385540314, −4.01279706048865905296986934191, −3.26219571471016544980217150176, −2.44603374903118612157641698027, −1.22830217318370386048665896755, 0,
1.22830217318370386048665896755, 2.44603374903118612157641698027, 3.26219571471016544980217150176, 4.01279706048865905296986934191, 4.71711643897163330818385540314, 5.70196145539737324099653241529, 6.37047528066041968073239420950, 6.91252972203290249423273095538, 7.72099804916880757403741733197
