L(s) = 1 | − 5-s + 3·7-s + 5·11-s − 13-s − 3·17-s − 8·19-s − 3·23-s + 25-s − 2·29-s − 10·31-s − 3·35-s − 3·37-s + 3·41-s − 10·43-s − 6·47-s + 2·49-s + 5·53-s − 5·55-s + 4·59-s − 7·61-s + 65-s + 12·67-s + 5·71-s + 2·73-s + 15·77-s − 13·79-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.13·7-s + 1.50·11-s − 0.277·13-s − 0.727·17-s − 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.371·29-s − 1.79·31-s − 0.507·35-s − 0.493·37-s + 0.468·41-s − 1.52·43-s − 0.875·47-s + 2/7·49-s + 0.686·53-s − 0.674·55-s + 0.520·59-s − 0.896·61-s + 0.124·65-s + 1.46·67-s + 0.593·71-s + 0.234·73-s + 1.70·77-s − 1.46·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.106015285253792308041943452369, −7.12300061082889949254379262330, −6.64969266894382826780695380646, −5.76412915318161576950950537772, −4.80203419334775113823930016566, −4.18589609170564321574894760747, −3.60230793452698877248812883980, −2.13766669817990869188295386398, −1.57342008865319396789694142328, 0,
1.57342008865319396789694142328, 2.13766669817990869188295386398, 3.60230793452698877248812883980, 4.18589609170564321574894760747, 4.80203419334775113823930016566, 5.76412915318161576950950537772, 6.64969266894382826780695380646, 7.12300061082889949254379262330, 8.106015285253792308041943452369