| L(s) = 1 | + 18·2-s − 81·3-s − 188·4-s + 1.53e3·5-s − 1.45e3·6-s − 1.26e4·8-s + 6.56e3·9-s + 2.75e4·10-s + 2.11e4·11-s + 1.52e4·12-s − 3.12e4·13-s − 1.23e5·15-s − 1.30e5·16-s + 2.79e5·17-s + 1.18e5·18-s − 1.44e5·19-s − 2.87e5·20-s + 3.80e5·22-s − 1.76e6·23-s + 1.02e6·24-s + 3.87e5·25-s − 5.61e5·26-s − 5.31e5·27-s + 4.69e6·29-s − 2.23e6·30-s + 3.69e5·31-s + 4.10e6·32-s + ⋯ |
| L(s) = 1 | + 0.795·2-s − 0.577·3-s − 0.367·4-s + 1.09·5-s − 0.459·6-s − 1.08·8-s + 1/3·9-s + 0.870·10-s + 0.435·11-s + 0.211·12-s − 0.303·13-s − 0.632·15-s − 0.497·16-s + 0.811·17-s + 0.265·18-s − 0.253·19-s − 0.401·20-s + 0.346·22-s − 1.31·23-s + 0.627·24-s + 0.198·25-s − 0.241·26-s − 0.192·27-s + 1.23·29-s − 0.502·30-s + 0.0717·31-s + 0.691·32-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
| \(L(5)\) |
\(\approx\) |
\(2.605673568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.605673568\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 9 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 306 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 21132 T + p^{9} T^{2} \) |
| 13 | \( 1 + 31214 T + p^{9} T^{2} \) |
| 17 | \( 1 - 279342 T + p^{9} T^{2} \) |
| 19 | \( 1 + 7580 p T + p^{9} T^{2} \) |
| 23 | \( 1 + 1763496 T + p^{9} T^{2} \) |
| 29 | \( 1 - 4692510 T + p^{9} T^{2} \) |
| 31 | \( 1 - 369088 T + p^{9} T^{2} \) |
| 37 | \( 1 - 9347078 T + p^{9} T^{2} \) |
| 41 | \( 1 - 7226838 T + p^{9} T^{2} \) |
| 43 | \( 1 + 23147476 T + p^{9} T^{2} \) |
| 47 | \( 1 + 22971888 T + p^{9} T^{2} \) |
| 53 | \( 1 - 78477174 T + p^{9} T^{2} \) |
| 59 | \( 1 - 20310660 T + p^{9} T^{2} \) |
| 61 | \( 1 - 179339938 T + p^{9} T^{2} \) |
| 67 | \( 1 - 274528388 T + p^{9} T^{2} \) |
| 71 | \( 1 + 36342648 T + p^{9} T^{2} \) |
| 73 | \( 1 - 247089526 T + p^{9} T^{2} \) |
| 79 | \( 1 - 191874800 T + p^{9} T^{2} \) |
| 83 | \( 1 - 276159276 T + p^{9} T^{2} \) |
| 89 | \( 1 - 678997350 T + p^{9} T^{2} \) |
| 97 | \( 1 - 567657502 T + p^{9} T^{2} \) |
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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
−11.66893362125094308399219152984, −10.14163769249970102283264469747, −9.597505473385119078088757990416, −8.266801387276214610948410223709, −6.56518012614549695989586812674, −5.80351834279439866821145449395, −4.92723606223869979590459173772, −3.75226265669756273155787955614, −2.24785870000832434099039753916, −0.76438343007198811956369318698,
0.76438343007198811956369318698, 2.24785870000832434099039753916, 3.75226265669756273155787955614, 4.92723606223869979590459173772, 5.80351834279439866821145449395, 6.56518012614549695989586812674, 8.266801387276214610948410223709, 9.597505473385119078088757990416, 10.14163769249970102283264469747, 11.66893362125094308399219152984
