L(s) = 1 | − 4·7-s − 4·11-s − 2·13-s − 6·17-s − 4·19-s + 2·29-s − 4·31-s − 2·37-s − 2·41-s − 4·43-s − 8·47-s + 9·49-s − 10·53-s + 4·59-s − 6·61-s − 4·67-s − 16·71-s + 6·73-s + 16·77-s − 4·79-s + 12·83-s − 10·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.371·29-s − 0.718·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.520·59-s − 0.768·61-s − 0.488·67-s − 1.89·71-s + 0.702·73-s + 1.82·77-s − 0.450·79-s + 1.31·83-s − 1.05·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.50058601687625, −16.09850222810448, −15.62227938750429, −15.11278203745670, −14.55863705845527, −13.61504323876422, −13.24316781095115, −12.85148325137776, −12.37442411186646, −11.61192741345699, −10.82577721334483, −10.44798086721231, −9.843932202764715, −9.259870263630899, −8.688852348670074, −7.994429340397551, −7.277594224205963, −6.571437356661767, −6.307659277764639, −5.362767649284238, −4.740027493498347, −3.989887907315521, −3.105774331002156, −2.617375081335161, −1.783942119495822, 0, 0,
1.783942119495822, 2.617375081335161, 3.105774331002156, 3.989887907315521, 4.740027493498347, 5.362767649284238, 6.307659277764639, 6.571437356661767, 7.277594224205963, 7.994429340397551, 8.688852348670074, 9.259870263630899, 9.843932202764715, 10.44798086721231, 10.82577721334483, 11.61192741345699, 12.37442411186646, 12.85148325137776, 13.24316781095115, 13.61504323876422, 14.55863705845527, 15.11278203745670, 15.62227938750429, 16.09850222810448, 16.50058601687625