| L(s) = 1 | − 24·2-s − 81·3-s + 64·4-s + 144·5-s + 1.94e3·6-s + 1.07e4·8-s + 6.56e3·9-s − 3.45e3·10-s − 1.50e4·11-s − 5.18e3·12-s + 1.51e5·13-s − 1.16e4·15-s − 2.90e5·16-s + 3.50e5·17-s − 1.57e5·18-s + 6.91e5·19-s + 9.21e3·20-s + 3.60e5·22-s + 8.92e5·23-s − 8.70e5·24-s − 1.93e6·25-s − 3.63e6·26-s − 5.31e5·27-s + 1.64e6·29-s + 2.79e5·30-s + 3.73e6·31-s + 1.47e6·32-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.103·5-s + 0.612·6-s + 0.928·8-s + 1/3·9-s − 0.109·10-s − 0.309·11-s − 0.0721·12-s + 1.47·13-s − 0.0594·15-s − 1.10·16-s + 1.01·17-s − 0.353·18-s + 1.21·19-s + 0.0128·20-s + 0.328·22-s + 0.664·23-s − 0.535·24-s − 0.989·25-s − 1.56·26-s − 0.192·27-s + 0.432·29-s + 0.0630·30-s + 0.726·31-s + 0.248·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.008087228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008087228\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{4} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 p^{3} T + p^{9} T^{2} \) |
| 5 | \( 1 - 144 T + p^{9} T^{2} \) |
| 11 | \( 1 + 15030 T + p^{9} T^{2} \) |
| 13 | \( 1 - 151486 T + p^{9} T^{2} \) |
| 17 | \( 1 - 350448 T + p^{9} T^{2} \) |
| 19 | \( 1 - 691108 T + p^{9} T^{2} \) |
| 23 | \( 1 - 892458 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1648518 T + p^{9} T^{2} \) |
| 31 | \( 1 - 3734296 T + p^{9} T^{2} \) |
| 37 | \( 1 + 11471902 T + p^{9} T^{2} \) |
| 41 | \( 1 + 13985724 T + p^{9} T^{2} \) |
| 43 | \( 1 - 16794524 T + p^{9} T^{2} \) |
| 47 | \( 1 - 14012052 T + p^{9} T^{2} \) |
| 53 | \( 1 + 97439910 T + p^{9} T^{2} \) |
| 59 | \( 1 + 110798304 T + p^{9} T^{2} \) |
| 61 | \( 1 - 93816682 T + p^{9} T^{2} \) |
| 67 | \( 1 + 122446456 T + p^{9} T^{2} \) |
| 71 | \( 1 - 206197398 T + p^{9} T^{2} \) |
| 73 | \( 1 + 250337558 T + p^{9} T^{2} \) |
| 79 | \( 1 + 38314852 T + p^{9} T^{2} \) |
| 83 | \( 1 - 514086924 T + p^{9} T^{2} \) |
| 89 | \( 1 - 1061294916 T + p^{9} T^{2} \) |
| 97 | \( 1 - 73841578 T + p^{9} T^{2} \) |
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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
−11.05268153332427583490819035675, −10.23850191390012048111835174277, −9.359715466161379675702334325674, −8.283856728288546561007168344061, −7.38807143255844215993796730993, −6.05650427627293763704307824438, −4.92539079814761226046898120550, −3.45049866559487882132880911763, −1.51126929943649263642632253398, −0.69257699918652788428062506000,
0.69257699918652788428062506000, 1.51126929943649263642632253398, 3.45049866559487882132880911763, 4.92539079814761226046898120550, 6.05650427627293763704307824438, 7.38807143255844215993796730993, 8.283856728288546561007168344061, 9.359715466161379675702334325674, 10.23850191390012048111835174277, 11.05268153332427583490819035675
