L(s) = 1 | + 32·2-s + 1.02e3·4-s + 1.17e4·5-s − 5.00e4·7-s + 3.27e4·8-s + 3.75e5·10-s + 5.31e5·11-s + 1.33e6·13-s − 1.60e6·14-s + 1.04e6·16-s + 5.10e6·17-s + 2.90e6·19-s + 1.20e7·20-s + 1.70e7·22-s − 3.05e7·23-s + 8.87e7·25-s + 4.26e7·26-s − 5.12e7·28-s + 7.70e7·29-s − 2.39e8·31-s + 3.35e7·32-s + 1.63e8·34-s − 5.86e8·35-s − 7.85e8·37-s + 9.28e7·38-s + 3.84e8·40-s − 4.11e8·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.67·5-s − 1.12·7-s + 0.353·8-s + 1.18·10-s + 0.994·11-s + 0.995·13-s − 0.795·14-s + 1/4·16-s + 0.872·17-s + 0.268·19-s + 0.839·20-s + 0.703·22-s − 0.991·23-s + 1.81·25-s + 0.703·26-s − 0.562·28-s + 0.697·29-s − 1.50·31-s + 0.176·32-s + 0.617·34-s − 1.88·35-s − 1.86·37-s + 0.190·38-s + 0.593·40-s − 0.554·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(3.481310618\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.481310618\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2346 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 7144 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 531420 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1332566 T + p^{11} T^{2} \) |
| 17 | \( 1 - 5109678 T + p^{11} T^{2} \) |
| 19 | \( 1 - 2901404 T + p^{11} T^{2} \) |
| 23 | \( 1 + 30597000 T + p^{11} T^{2} \) |
| 29 | \( 1 - 77006634 T + p^{11} T^{2} \) |
| 31 | \( 1 + 239418352 T + p^{11} T^{2} \) |
| 37 | \( 1 + 785041666 T + p^{11} T^{2} \) |
| 41 | \( 1 + 411252954 T + p^{11} T^{2} \) |
| 43 | \( 1 - 351233348 T + p^{11} T^{2} \) |
| 47 | \( 1 + 95821680 T + p^{11} T^{2} \) |
| 53 | \( 1 - 1465857378 T + p^{11} T^{2} \) |
| 59 | \( 1 + 5621152020 T + p^{11} T^{2} \) |
| 61 | \( 1 + 10473587770 T + p^{11} T^{2} \) |
| 67 | \( 1 - 4515307532 T + p^{11} T^{2} \) |
| 71 | \( 1 - 8509579560 T + p^{11} T^{2} \) |
| 73 | \( 1 - 2012496986 T + p^{11} T^{2} \) |
| 79 | \( 1 + 22238409568 T + p^{11} T^{2} \) |
| 83 | \( 1 + 6328647516 T + p^{11} T^{2} \) |
| 89 | \( 1 - 50123706678 T + p^{11} T^{2} \) |
| 97 | \( 1 - 94805961314 T + p^{11} T^{2} \) |
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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
−16.05450114493974296068423188073, −14.26770594204554709843088145895, −13.51628284255213760889327899401, −12.31015133449301585407918873312, −10.36840032055032392221638585038, −9.224191660655688862679435719552, −6.57959580553579358598933389693, −5.68971361281167691539568307649, −3.41771560822878169631254205535, −1.60457919752433608316953414142,
1.60457919752433608316953414142, 3.41771560822878169631254205535, 5.68971361281167691539568307649, 6.57959580553579358598933389693, 9.224191660655688862679435719552, 10.36840032055032392221638585038, 12.31015133449301585407918873312, 13.51628284255213760889327899401, 14.26770594204554709843088145895, 16.05450114493974296068423188073