| L(s) = 1 | − 66.4·3-s − 157.·5-s + 343·7-s + 2.22e3·9-s + 6.20e3·11-s + 5.38e3·13-s + 1.04e4·15-s − 1.09e4·17-s + 1.17e4·19-s − 2.27e4·21-s + 1.06e5·23-s − 5.34e4·25-s − 2.36e3·27-s − 5.15e4·29-s − 2.47e5·31-s − 4.12e5·33-s − 5.39e4·35-s + 4.33e5·37-s − 3.57e5·39-s + 3.22e5·41-s + 8.78e5·43-s − 3.49e5·45-s + 6.55e5·47-s + 1.17e5·49-s + 7.30e5·51-s − 4.44e5·53-s − 9.76e5·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 0.562·5-s + 0.377·7-s + 1.01·9-s + 1.40·11-s + 0.679·13-s + 0.798·15-s − 0.542·17-s + 0.391·19-s − 0.536·21-s + 1.81·23-s − 0.683·25-s − 0.0231·27-s − 0.392·29-s − 1.49·31-s − 1.99·33-s − 0.212·35-s + 1.40·37-s − 0.964·39-s + 0.731·41-s + 1.68·43-s − 0.571·45-s + 0.920·47-s + 0.142·49-s + 0.770·51-s − 0.410·53-s − 0.791·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.012637088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012637088\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 + 66.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 157.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.20e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.17e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.15e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.47e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.22e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.78e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.55e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.44e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.92e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.58e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.10e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.86e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.71e5T + 8.07e13T^{2} \) |
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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
−15.92452442851786385857106676770, −14.56400510428925775983768184847, −12.82254298316349620197232123431, −11.49644384530215454587305652872, −11.07652909017344734539039550225, −9.096038049657601570719226328952, −7.14169875331353586097768383755, −5.77783527037691559786222184223, −4.16526462610390347184493289631, −0.945840617285120593439029079535,
0.945840617285120593439029079535, 4.16526462610390347184493289631, 5.77783527037691559786222184223, 7.14169875331353586097768383755, 9.096038049657601570719226328952, 11.07652909017344734539039550225, 11.49644384530215454587305652872, 12.82254298316349620197232123431, 14.56400510428925775983768184847, 15.92452442851786385857106676770
