| L(s) = 1 | + 2·5-s + 2·7-s − 4·11-s + 13-s − 6·19-s − 4·23-s − 25-s + 8·29-s + 2·31-s + 4·35-s − 6·37-s + 6·41-s − 8·43-s − 8·47-s − 3·49-s − 12·53-s − 8·55-s + 4·59-s − 10·61-s + 2·65-s − 2·67-s + 16·71-s + 14·73-s − 8·77-s + 4·79-s − 12·83-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.20·11-s + 0.277·13-s − 1.37·19-s − 0.834·23-s − 1/5·25-s + 1.48·29-s + 0.359·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s − 3/7·49-s − 1.64·53-s − 1.07·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.244·67-s + 1.89·71-s + 1.63·73-s − 0.911·77-s + 0.450·79-s − 1.31·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.82885782269489685929763316343, −6.54658570849644716266634653386, −6.31902631650893558385157451977, −5.29958688295089777719730657827, −4.89846206181494562517735323880, −4.04504924809123623904528509602, −2.94012618851587332451134020886, −2.16841436964911272483029187855, −1.50688699187961671245243796764, 0,
1.50688699187961671245243796764, 2.16841436964911272483029187855, 2.94012618851587332451134020886, 4.04504924809123623904528509602, 4.89846206181494562517735323880, 5.29958688295089777719730657827, 6.31902631650893558385157451977, 6.54658570849644716266634653386, 7.82885782269489685929763316343
