| L(s) = 1 | + 2-s − 2·3-s − 7·4-s + 16·5-s − 2·6-s + 7·7-s − 15·8-s − 23·9-s + 16·10-s + 14·12-s − 28·13-s + 7·14-s − 32·15-s + 41·16-s − 54·17-s − 23·18-s + 110·19-s − 112·20-s − 14·21-s + 48·23-s + 30·24-s + 131·25-s − 28·26-s + 100·27-s − 49·28-s + 110·29-s − 32·30-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.384·3-s − 7/8·4-s + 1.43·5-s − 0.136·6-s + 0.377·7-s − 0.662·8-s − 0.851·9-s + 0.505·10-s + 0.336·12-s − 0.597·13-s + 0.133·14-s − 0.550·15-s + 0.640·16-s − 0.770·17-s − 0.301·18-s + 1.32·19-s − 1.25·20-s − 0.145·21-s + 0.435·23-s + 0.255·24-s + 1.04·25-s − 0.211·26-s + 0.712·27-s − 0.330·28-s + 0.704·29-s − 0.194·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.963696696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963696696\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 110 T + p^{3} T^{2} \) |
| 31 | \( 1 - 12 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 182 T + p^{3} T^{2} \) |
| 43 | \( 1 + 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 810 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 702 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 - 730 T + p^{3} T^{2} \) |
| 97 | \( 1 - 294 T + p^{3} T^{2} \) |
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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
−9.787081795200737663507816397655, −9.028228598826557838607406890545, −8.377770933314236780038474866280, −7.01147939019497066438910267258, −6.01365907896678875018612017917, −5.28285927970685486895338944527, −4.83481037533302843201193190111, −3.31918745569909271743506679309, −2.21345251215478291384990864744, −0.75316267854140113947099853893,
0.75316267854140113947099853893, 2.21345251215478291384990864744, 3.31918745569909271743506679309, 4.83481037533302843201193190111, 5.28285927970685486895338944527, 6.01365907896678875018612017917, 7.01147939019497066438910267258, 8.377770933314236780038474866280, 9.028228598826557838607406890545, 9.787081795200737663507816397655
