| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 2·11-s + 3·13-s + 15-s − 2·17-s + 5·19-s − 3·21-s + 4·23-s + 25-s − 27-s − 29-s − 2·33-s − 3·35-s − 3·39-s − 12·41-s − 4·43-s − 45-s − 6·47-s + 2·49-s + 2·51-s + 12·53-s − 2·55-s − 5·57-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.832·13-s + 0.258·15-s − 0.485·17-s + 1.14·19-s − 0.654·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s − 0.348·33-s − 0.507·35-s − 0.480·39-s − 1.87·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 2/7·49-s + 0.280·51-s + 1.64·53-s − 0.269·55-s − 0.662·57-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.91567885921741, −12.08084416128072, −11.86346385038761, −11.46349840054988, −11.13270624809667, −10.77399149304292, −10.14101486683264, −9.709441539803488, −9.103940825914454, −8.627830476068031, −8.245317815222336, −7.843424580437769, −7.120124473634195, −6.849947417384736, −6.420686341389058, −5.666903032293594, −5.174018317198219, −4.967887885019537, −4.308837802804650, −3.717886975660092, −3.425789341442035, −2.598613913632997, −1.849191045097021, −1.301369814492647, −0.9174267293619499, 0,
0.9174267293619499, 1.301369814492647, 1.849191045097021, 2.598613913632997, 3.425789341442035, 3.717886975660092, 4.308837802804650, 4.967887885019537, 5.174018317198219, 5.666903032293594, 6.420686341389058, 6.849947417384736, 7.120124473634195, 7.843424580437769, 8.245317815222336, 8.627830476068031, 9.103940825914454, 9.709441539803488, 10.14101486683264, 10.77399149304292, 11.13270624809667, 11.46349840054988, 11.86346385038761, 12.08084416128072, 12.91567885921741
