| L(s) = 1 | + 3-s + 5-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s + 25-s + 27-s − 6·29-s − 4·31-s + 2·37-s − 2·39-s − 6·41-s − 8·43-s + 45-s − 12·47-s + 6·51-s + 6·53-s − 4·57-s − 12·59-s − 2·61-s − 2·65-s − 8·67-s − 14·73-s + 75-s + 16·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.63·73-s + 0.115·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.65923486397524, −16.36636522893473, −15.26951460723723, −14.88168406505266, −14.59780154542738, −13.83365536132892, −13.30589981083666, −12.79977899153455, −12.17278071016755, −11.62756652869789, −10.73660559095293, −10.28122212767041, −9.605093206856188, −9.248561191636635, −8.396791088790065, −7.907116623875908, −7.256162742972956, −6.586220244527530, −5.823254019156098, −5.182624051924949, −4.474606958703111, −3.539659067778729, −3.039946646525370, −2.035155177139918, −1.454593903281767, 0,
1.454593903281767, 2.035155177139918, 3.039946646525370, 3.539659067778729, 4.474606958703111, 5.182624051924949, 5.823254019156098, 6.586220244527530, 7.256162742972956, 7.907116623875908, 8.396791088790065, 9.248561191636635, 9.605093206856188, 10.28122212767041, 10.73660559095293, 11.62756652869789, 12.17278071016755, 12.79977899153455, 13.30589981083666, 13.83365536132892, 14.59780154542738, 14.88168406505266, 15.26951460723723, 16.36636522893473, 16.65923486397524
