| L(s) = 1 | + 2·3-s + 9-s + 2·11-s − 6·19-s + 6·23-s − 4·27-s + 6·29-s + 6·31-s + 4·33-s − 6·37-s − 8·41-s + 6·43-s − 8·47-s − 7·49-s − 12·53-s − 12·57-s − 2·59-s − 6·61-s + 12·67-s + 12·69-s − 2·71-s + 6·73-s − 11·81-s + 4·83-s + 12·87-s − 8·89-s + 12·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.603·11-s − 1.37·19-s + 1.25·23-s − 0.769·27-s + 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.986·37-s − 1.24·41-s + 0.914·43-s − 1.16·47-s − 49-s − 1.64·53-s − 1.58·57-s − 0.260·59-s − 0.768·61-s + 1.46·67-s + 1.44·69-s − 0.237·71-s + 0.702·73-s − 1.22·81-s + 0.439·83-s + 1.28·87-s − 0.847·89-s + 1.24·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.00960423132083, −12.69181908748606, −12.14953228265620, −11.66795218360207, −11.06390729495350, −10.80063820954121, −10.06770762372084, −9.759026461414146, −9.199404629237336, −8.749714951437987, −8.418337005860764, −8.070086237715093, −7.509840279483511, −6.769545698299659, −6.565388310691551, −6.081515205600838, −5.262456252332137, −4.654464022721500, −4.443989102899532, −3.533847897735914, −3.282210388978640, −2.774957204282469, −2.078301043122864, −1.658154450254525, −0.8926437861228895, 0,
0.8926437861228895, 1.658154450254525, 2.078301043122864, 2.774957204282469, 3.282210388978640, 3.533847897735914, 4.443989102899532, 4.654464022721500, 5.262456252332137, 6.081515205600838, 6.565388310691551, 6.769545698299659, 7.509840279483511, 8.070086237715093, 8.418337005860764, 8.749714951437987, 9.199404629237336, 9.759026461414146, 10.06770762372084, 10.80063820954121, 11.06390729495350, 11.66795218360207, 12.14953228265620, 12.69181908748606, 13.00960423132083
