L(s) = 1 | + 3-s − 7-s + 9-s + 7·13-s + 4·17-s − 3·19-s − 21-s − 6·23-s + 27-s − 8·29-s + 3·31-s − 11·37-s + 7·39-s − 6·41-s − 43-s + 4·47-s + 49-s + 4·51-s − 6·53-s − 3·57-s − 6·59-s − 13·61-s − 63-s + 5·67-s − 6·69-s + 6·71-s + 3·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.94·13-s + 0.970·17-s − 0.688·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.48·29-s + 0.538·31-s − 1.80·37-s + 1.12·39-s − 0.937·41-s − 0.152·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s − 0.397·57-s − 0.781·59-s − 1.66·61-s − 0.125·63-s + 0.610·67-s − 0.722·69-s + 0.712·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.327010947\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.327010947\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.85979780701165, −12.30181952141847, −12.15206340039390, −11.37350992652658, −10.91064724687997, −10.56564524645048, −10.00388266277545, −9.621054935564640, −9.018893301504478, −8.603559351273410, −8.269554260718478, −7.721141891802926, −7.305087741906256, −6.498456011078257, −6.302332310910821, −5.731069118732690, −5.250496826704931, −4.485917510551780, −3.899891228395947, −3.452623277241407, −3.284161379710503, −2.337685502799783, −1.697501101939126, −1.371334142135616, −0.3883345116660467,
0.3883345116660467, 1.371334142135616, 1.697501101939126, 2.337685502799783, 3.284161379710503, 3.452623277241407, 3.899891228395947, 4.485917510551780, 5.250496826704931, 5.731069118732690, 6.302332310910821, 6.498456011078257, 7.305087741906256, 7.721141891802926, 8.269554260718478, 8.603559351273410, 9.018893301504478, 9.621054935564640, 10.00388266277545, 10.56564524645048, 10.91064724687997, 11.37350992652658, 12.15206340039390, 12.30181952141847, 12.85979780701165