| L(s) = 1 | + 16·2-s + 256·4-s − 2.52e3·5-s + 7.03e3·7-s + 4.09e3·8-s − 4.04e4·10-s − 3.94e4·11-s + 4.77e4·13-s + 1.12e5·14-s + 6.55e4·16-s − 4.20e5·17-s + 4.86e5·19-s − 6.46e5·20-s − 6.30e5·22-s − 1.12e6·23-s + 4.42e6·25-s + 7.64e5·26-s + 1.80e6·28-s − 3.19e6·29-s − 3.55e6·31-s + 1.04e6·32-s − 6.72e6·34-s − 1.77e7·35-s − 2.21e6·37-s + 7.78e6·38-s − 1.03e7·40-s + 2.87e7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.80·5-s + 1.10·7-s + 0.353·8-s − 1.27·10-s − 0.811·11-s + 0.464·13-s + 0.783·14-s + 0.250·16-s − 1.22·17-s + 0.856·19-s − 0.903·20-s − 0.573·22-s − 0.840·23-s + 2.26·25-s + 0.328·26-s + 0.553·28-s − 0.838·29-s − 0.691·31-s + 0.176·32-s − 0.862·34-s − 2.00·35-s − 0.194·37-s + 0.605·38-s − 0.639·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.355782363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.355782363\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.52e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.03e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.94e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.77e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.20e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.86e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.19e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.55e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.21e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.87e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.02e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.38e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.48e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.27e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.24e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.35e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.55e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 7.34e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.66e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.04e8T + 7.60e17T^{2} \) |
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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
−11.26110119523196157305562737140, −10.77407734001840915833584637406, −8.807081075868433196450137881582, −7.80690307469841129067489338854, −7.27368894286356212120099989309, −5.57803554662277812252137777686, −4.45258113023757639277035334396, −3.77331587641911901115529315686, −2.33124411361533269658110687917, −0.68550034877451641189041913662,
0.68550034877451641189041913662, 2.33124411361533269658110687917, 3.77331587641911901115529315686, 4.45258113023757639277035334396, 5.57803554662277812252137777686, 7.27368894286356212120099989309, 7.80690307469841129067489338854, 8.807081075868433196450137881582, 10.77407734001840915833584637406, 11.26110119523196157305562737140
