| L(s) = 1 | + 3·3-s − 8·4-s − 2·7-s + 9·9-s − 11·11-s − 24·12-s + 22·13-s + 64·16-s − 72·17-s + 122·19-s − 6·21-s − 72·23-s + 27·27-s + 16·28-s + 96·29-s − 112·31-s − 33·33-s − 72·36-s − 266·37-s + 66·39-s − 96·41-s + 382·43-s + 88·44-s − 360·47-s + 192·48-s − 339·49-s − 216·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.107·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 0.469·13-s + 16-s − 1.02·17-s + 1.47·19-s − 0.0623·21-s − 0.652·23-s + 0.192·27-s + 0.107·28-s + 0.614·29-s − 0.648·31-s − 0.174·33-s − 1/3·36-s − 1.18·37-s + 0.270·39-s − 0.365·41-s + 1.35·43-s + 0.301·44-s − 1.11·47-s + 0.577·48-s − 0.988·49-s − 0.593·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 122 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 96 T + p^{3} T^{2} \) |
| 43 | \( 1 - 382 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 430 T + p^{3} T^{2} \) |
| 67 | \( 1 + 380 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 706 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1068 T + p^{3} T^{2} \) |
| 89 | \( 1 + 6 T + p^{3} T^{2} \) |
| 97 | \( 1 + 686 T + p^{3} T^{2} \) |
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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
−9.363252259449192128624865426005, −8.610521330296451039658275045690, −7.928454194132104779784890868792, −6.94268162317599509189957641743, −5.74437026653176259668032328609, −4.81776315307113769974819199589, −3.87496281931032794610050345834, −2.95865112366885103391089029523, −1.46089361123687494602973421762, 0,
1.46089361123687494602973421762, 2.95865112366885103391089029523, 3.87496281931032794610050345834, 4.81776315307113769974819199589, 5.74437026653176259668032328609, 6.94268162317599509189957641743, 7.928454194132104779784890868792, 8.610521330296451039658275045690, 9.363252259449192128624865426005
