| L(s) = 1 | − 3-s − 5-s + 9-s + 3·11-s + 13-s + 15-s − 4·17-s − 19-s + 5·23-s + 25-s − 27-s + 4·29-s + 2·31-s − 3·33-s − 3·37-s − 39-s + 3·41-s − 4·43-s − 45-s − 9·47-s + 4·51-s + 3·53-s − 3·55-s + 57-s − 12·59-s − 65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s − 0.970·17-s − 0.229·19-s + 1.04·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.359·31-s − 0.522·33-s − 0.493·37-s − 0.160·39-s + 0.468·41-s − 0.609·43-s − 0.149·45-s − 1.31·47-s + 0.560·51-s + 0.412·53-s − 0.404·55-s + 0.132·57-s − 1.56·59-s − 0.124·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.74976129441650, −15.20105514213738, −14.77940066201212, −14.09552155802711, −13.49768778379398, −12.99626962749809, −12.38475323769240, −11.85648207106254, −11.40582312541478, −10.87491406605342, −10.43054751012855, −9.618574251105378, −9.086849204721928, −8.545473410159158, −7.950067745429917, −7.123777063215332, −6.638343883246898, −6.280001493140100, −5.404578879260377, −4.693451989787916, −4.287870254649834, −3.500716749449960, −2.796203032505857, −1.752126138730818, −1.015245237481568, 0,
1.015245237481568, 1.752126138730818, 2.796203032505857, 3.500716749449960, 4.287870254649834, 4.693451989787916, 5.404578879260377, 6.280001493140100, 6.638343883246898, 7.123777063215332, 7.950067745429917, 8.545473410159158, 9.086849204721928, 9.618574251105378, 10.43054751012855, 10.87491406605342, 11.40582312541478, 11.85648207106254, 12.38475323769240, 12.99626962749809, 13.49768778379398, 14.09552155802711, 14.77940066201212, 15.20105514213738, 15.74976129441650
