L(s) = 1 | − 3-s + 4·7-s + 9-s + 13-s − 5·19-s − 4·21-s − 27-s + 3·29-s + 4·31-s − 7·37-s − 39-s + 3·41-s − 2·43-s − 9·47-s + 9·49-s + 9·53-s + 5·57-s − 6·59-s + 8·61-s + 4·63-s − 5·67-s + 3·71-s − 4·73-s − 11·79-s + 81-s + 6·83-s − 3·87-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 1.14·19-s − 0.872·21-s − 0.192·27-s + 0.557·29-s + 0.718·31-s − 1.15·37-s − 0.160·39-s + 0.468·41-s − 0.304·43-s − 1.31·47-s + 9/7·49-s + 1.23·53-s + 0.662·57-s − 0.781·59-s + 1.02·61-s + 0.503·63-s − 0.610·67-s + 0.356·71-s − 0.468·73-s − 1.23·79-s + 1/9·81-s + 0.658·83-s − 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.078615714\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.078615714\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.94920269128663, −15.45325272577533, −14.88441704000454, −14.37249636616361, −13.87963410817147, −13.14893545163824, −12.64960428509959, −11.82951164985675, −11.60754260367119, −10.97785764450725, −10.39312661658442, −10.03760712440474, −8.903513450257530, −8.577286139197537, −7.951219772231055, −7.328355507931167, −6.584528956492519, −6.022866053666940, −5.243560838031910, −4.704592672400860, −4.227085676882818, −3.296750194555910, −2.196705947795707, −1.616005592490480, −0.6654524303542898,
0.6654524303542898, 1.616005592490480, 2.196705947795707, 3.296750194555910, 4.227085676882818, 4.704592672400860, 5.243560838031910, 6.022866053666940, 6.584528956492519, 7.328355507931167, 7.951219772231055, 8.577286139197537, 8.903513450257530, 10.03760712440474, 10.39312661658442, 10.97785764450725, 11.60754260367119, 11.82951164985675, 12.64960428509959, 13.14893545163824, 13.87963410817147, 14.37249636616361, 14.88441704000454, 15.45325272577533, 15.94920269128663