| L(s) = 1 | + 5-s − 4·11-s + 4·13-s − 4·17-s + 4·19-s − 23-s + 25-s − 6·29-s + 4·31-s − 2·37-s − 4·43-s − 7·49-s − 6·53-s − 4·55-s − 6·59-s − 4·61-s + 4·65-s + 4·67-s − 4·71-s − 2·73-s + 10·79-s + 18·83-s − 4·85-s − 2·89-s + 4·95-s + 2·97-s + 10·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 1.10·13-s − 0.970·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.609·43-s − 49-s − 0.824·53-s − 0.539·55-s − 0.781·59-s − 0.512·61-s + 0.496·65-s + 0.488·67-s − 0.474·71-s − 0.234·73-s + 1.12·79-s + 1.97·83-s − 0.433·85-s − 0.211·89-s + 0.410·95-s + 0.203·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.64321127360273118771531552055, −6.60421657520843199521833712249, −6.17002273576854128317567990156, −5.30551781363170447665809521831, −4.83799266903610436392152565541, −3.80213001960659387441815523893, −3.07158251592705580707875149849, −2.21417025013198550178231962412, −1.33478221883967991045799498918, 0,
1.33478221883967991045799498918, 2.21417025013198550178231962412, 3.07158251592705580707875149849, 3.80213001960659387441815523893, 4.83799266903610436392152565541, 5.30551781363170447665809521831, 6.17002273576854128317567990156, 6.60421657520843199521833712249, 7.64321127360273118771531552055
