| L(s) = 1 | − 2-s − 7·4-s + 6·7-s + 15·8-s − 47·11-s + 5·13-s − 6·14-s + 41·16-s + 131·17-s − 56·19-s + 47·22-s − 3·23-s − 5·26-s − 42·28-s − 157·29-s + 225·31-s − 161·32-s − 131·34-s + 70·37-s + 56·38-s + 140·41-s − 397·43-s + 329·44-s + 3·46-s + 347·47-s − 307·49-s − 35·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.323·7-s + 0.662·8-s − 1.28·11-s + 0.106·13-s − 0.114·14-s + 0.640·16-s + 1.86·17-s − 0.676·19-s + 0.455·22-s − 0.0271·23-s − 0.0377·26-s − 0.283·28-s − 1.00·29-s + 1.30·31-s − 0.889·32-s − 0.660·34-s + 0.311·37-s + 0.239·38-s + 0.533·41-s − 1.40·43-s + 1.12·44-s + 0.00961·46-s + 1.07·47-s − 0.895·49-s − 0.0933·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 47 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 131 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 T + p^{3} T^{2} \) |
| 29 | \( 1 + 157 T + p^{3} T^{2} \) |
| 31 | \( 1 - 225 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 140 T + p^{3} T^{2} \) |
| 43 | \( 1 + 397 T + p^{3} T^{2} \) |
| 47 | \( 1 - 347 T + p^{3} T^{2} \) |
| 53 | \( 1 + 4 T + p^{3} T^{2} \) |
| 59 | \( 1 - 748 T + p^{3} T^{2} \) |
| 61 | \( 1 + 338 T + p^{3} T^{2} \) |
| 67 | \( 1 + 492 T + p^{3} T^{2} \) |
| 71 | \( 1 - 32 T + p^{3} T^{2} \) |
| 73 | \( 1 + 970 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1257 T + p^{3} T^{2} \) |
| 83 | \( 1 - 102 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 + 974 T + p^{3} T^{2} \) |
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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
−9.888017667925036055406293097765, −8.664675315631004916339990253944, −8.042704009743580861437420554675, −7.37674769738414007270718376546, −5.82816170667632843569602242566, −5.13169194601057253552778827397, −4.12069395305033916193114010103, −2.89776507464665104590265960988, −1.32478632933604666724255491577, 0,
1.32478632933604666724255491577, 2.89776507464665104590265960988, 4.12069395305033916193114010103, 5.13169194601057253552778827397, 5.82816170667632843569602242566, 7.37674769738414007270718376546, 8.042704009743580861437420554675, 8.664675315631004916339990253944, 9.888017667925036055406293097765
