| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 2·11-s − 13-s − 15-s + 17-s − 2·19-s + 2·21-s + 6·23-s + 25-s + 27-s + 5·29-s − 3·31-s − 2·33-s − 2·35-s + 4·37-s − 39-s + 10·41-s − 10·43-s − 45-s − 12·47-s − 3·49-s + 51-s − 3·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s − 0.538·31-s − 0.348·33-s − 0.338·35-s + 0.657·37-s − 0.160·39-s + 1.56·41-s − 1.52·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.140·51-s − 0.412·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−13.29763521404733, −12.74251032390885, −12.44908041404311, −11.78090792220975, −11.35621120107022, −10.87476468167203, −10.58120013962919, −9.833323931715581, −9.540563866909469, −8.907703853878710, −8.366989868581561, −8.121528676415059, −7.594902042772712, −7.219844164055194, −6.544540953328130, −6.147957711544069, −5.196281351783744, −4.975938357760950, −4.500288003226893, −3.880198123764030, −3.185285152918591, −2.845881278868585, −2.162684639454815, −1.535010572120421, −0.8836209158650331, 0,
0.8836209158650331, 1.535010572120421, 2.162684639454815, 2.845881278868585, 3.185285152918591, 3.880198123764030, 4.500288003226893, 4.975938357760950, 5.196281351783744, 6.147957711544069, 6.544540953328130, 7.219844164055194, 7.594902042772712, 8.121528676415059, 8.366989868581561, 8.907703853878710, 9.540563866909469, 9.833323931715581, 10.58120013962919, 10.87476468167203, 11.35621120107022, 11.78090792220975, 12.44908041404311, 12.74251032390885, 13.29763521404733
