| L(s) = 1 | − 3-s − 3·5-s + 9-s − 5·13-s + 3·15-s + 4·17-s − 19-s − 2·23-s + 4·25-s − 27-s − 5·29-s + 4·31-s − 3·37-s + 5·39-s + 6·41-s − 4·43-s − 3·45-s − 13·47-s − 4·51-s − 6·53-s + 57-s − 9·59-s − 2·61-s + 15·65-s + 5·67-s + 2·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.38·13-s + 0.774·15-s + 0.970·17-s − 0.229·19-s − 0.417·23-s + 4/5·25-s − 0.192·27-s − 0.928·29-s + 0.718·31-s − 0.493·37-s + 0.800·39-s + 0.937·41-s − 0.609·43-s − 0.447·45-s − 1.89·47-s − 0.560·51-s − 0.824·53-s + 0.132·57-s − 1.17·59-s − 0.256·61-s + 1.86·65-s + 0.610·67-s + 0.240·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.59600456387318, −12.93269822375567, −12.52371277798878, −12.10447650067089, −11.85669539591439, −11.23025564244752, −10.99388462625301, −10.20847789762585, −9.874379297856186, −9.423435198768607, −8.722187060985792, −8.009052293962955, −7.797359395982888, −7.389980824805744, −6.766588541574771, −6.309566155695253, −5.564938225508045, −5.118944198220380, −4.507994182878425, −4.213823787381314, −3.352443605435186, −3.113883947918479, −2.157818648191972, −1.498267307149852, −0.5557916728386336, 0,
0.5557916728386336, 1.498267307149852, 2.157818648191972, 3.113883947918479, 3.352443605435186, 4.213823787381314, 4.507994182878425, 5.118944198220380, 5.564938225508045, 6.309566155695253, 6.766588541574771, 7.389980824805744, 7.797359395982888, 8.009052293962955, 8.722187060985792, 9.423435198768607, 9.874379297856186, 10.20847789762585, 10.99388462625301, 11.23025564244752, 11.85669539591439, 12.10447650067089, 12.52371277798878, 12.93269822375567, 13.59600456387318
