| L(s) = 1 | + 2-s + 3·3-s − 7·4-s + 3·6-s + 26·7-s − 15·8-s + 9·9-s + 11·11-s − 21·12-s + 32·13-s + 26·14-s + 41·16-s − 74·17-s + 9·18-s − 60·19-s + 78·21-s + 11·22-s + 182·23-s − 45·24-s + 32·26-s + 27·27-s − 182·28-s − 90·29-s − 8·31-s + 161·32-s + 33·33-s − 74·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.204·6-s + 1.40·7-s − 0.662·8-s + 1/3·9-s + 0.301·11-s − 0.505·12-s + 0.682·13-s + 0.496·14-s + 0.640·16-s − 1.05·17-s + 0.117·18-s − 0.724·19-s + 0.810·21-s + 0.106·22-s + 1.64·23-s − 0.382·24-s + 0.241·26-s + 0.192·27-s − 1.22·28-s − 0.576·29-s − 0.0463·31-s + 0.889·32-s + 0.174·33-s − 0.373·34-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
| \(L(2)\) |
\(\approx\) |
\(2.982211532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.982211532\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 182 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 66 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 408 T + p^{3} T^{2} \) |
| 47 | \( 1 - 506 T + p^{3} T^{2} \) |
| 53 | \( 1 + 348 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 132 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 - 762 T + p^{3} T^{2} \) |
| 73 | \( 1 - 542 T + p^{3} T^{2} \) |
| 79 | \( 1 + 550 T + p^{3} T^{2} \) |
| 83 | \( 1 - 132 T + p^{3} T^{2} \) |
| 89 | \( 1 - 570 T + p^{3} T^{2} \) |
| 97 | \( 1 + 14 T + p^{3} T^{2} \) |
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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
−9.583345757957409493349995630346, −8.793165414019917646668046376778, −8.417628688043093599676955650668, −7.42047833014835507355357172359, −6.26430233038168183949653502064, −5.08760325235589785539595274130, −4.45317900765150053239862040574, −3.58325798115085777687166212781, −2.20059608316522608129176369566, −0.941720156224116459450854943026,
0.941720156224116459450854943026, 2.20059608316522608129176369566, 3.58325798115085777687166212781, 4.45317900765150053239862040574, 5.08760325235589785539595274130, 6.26430233038168183949653502064, 7.42047833014835507355357172359, 8.417628688043093599676955650668, 8.793165414019917646668046376778, 9.583345757957409493349995630346
