| L(s) = 1 | + 5-s + 4·7-s − 4·11-s + 4·13-s − 4·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s + 4·31-s + 4·35-s + 2·37-s − 10·41-s + 43-s + 4·47-s + 9·49-s + 2·53-s − 4·55-s − 12·59-s + 4·65-s − 8·67-s − 12·71-s − 14·73-s − 16·77-s − 8·79-s + 6·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.20·11-s + 1.10·13-s − 0.970·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s − 1.56·41-s + 0.152·43-s + 0.583·47-s + 9/7·49-s + 0.274·53-s − 0.539·55-s − 1.56·59-s + 0.496·65-s − 0.977·67-s − 1.42·71-s − 1.63·73-s − 1.82·77-s − 0.900·79-s + 0.658·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.38051325301860, −14.82711768633896, −14.44103250115402, −13.52451174666502, −13.30819925287600, −13.04160471982081, −12.08567165085941, −11.52368901145783, −10.97265485238109, −10.63881458531797, −10.30503455742277, −9.093592006945853, −8.963261274437155, −8.264187848849934, −7.856766182821188, −7.139938141099991, −6.548239288904613, −5.758455459159680, −5.339942111119706, −4.594983342024729, −4.310604716839754, −3.196199217105689, −2.555422519113520, −1.776843263210540, −1.255727938678685, 0,
1.255727938678685, 1.776843263210540, 2.555422519113520, 3.196199217105689, 4.310604716839754, 4.594983342024729, 5.339942111119706, 5.758455459159680, 6.548239288904613, 7.139938141099991, 7.856766182821188, 8.264187848849934, 8.963261274437155, 9.093592006945853, 10.30503455742277, 10.63881458531797, 10.97265485238109, 11.52368901145783, 12.08567165085941, 13.04160471982081, 13.30819925287600, 13.52451174666502, 14.44103250115402, 14.82711768633896, 15.38051325301860
