| L(s) = 1 | + 3·3-s − 2·5-s − 3·7-s + 6·9-s − 6·11-s − 5·13-s − 6·15-s − 19-s − 9·21-s + 5·23-s − 25-s + 9·27-s + 2·29-s − 5·31-s − 18·33-s + 6·35-s + 2·37-s − 15·39-s − 10·41-s − 43-s − 12·45-s + 47-s + 2·49-s + 6·53-s + 12·55-s − 3·57-s + 2·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s − 1.13·7-s + 2·9-s − 1.80·11-s − 1.38·13-s − 1.54·15-s − 0.229·19-s − 1.96·21-s + 1.04·23-s − 1/5·25-s + 1.73·27-s + 0.371·29-s − 0.898·31-s − 3.13·33-s + 1.01·35-s + 0.328·37-s − 2.40·39-s − 1.56·41-s − 0.152·43-s − 1.78·45-s + 0.145·47-s + 2/7·49-s + 0.824·53-s + 1.61·55-s − 0.397·57-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−12.83643358176674, −12.71488304375990, −12.03037525663718, −11.59913527206522, −10.84292169980134, −10.31695330355248, −10.06703776427735, −9.621635737884444, −9.128131058364961, −8.647129441861145, −8.243279266435460, −7.717645506483216, −7.451519851652322, −7.048354322534219, −6.607958216908488, −5.715027792237298, −5.125011966279813, −4.730971429650113, −4.067683022429273, −3.587431472131495, −3.012299793093073, −2.837344494874558, −2.291056470364580, −1.723510130687931, −0.5754880343968550, 0,
0.5754880343968550, 1.723510130687931, 2.291056470364580, 2.837344494874558, 3.012299793093073, 3.587431472131495, 4.067683022429273, 4.730971429650113, 5.125011966279813, 5.715027792237298, 6.607958216908488, 7.048354322534219, 7.451519851652322, 7.717645506483216, 8.243279266435460, 8.647129441861145, 9.128131058364961, 9.621635737884444, 10.06703776427735, 10.31695330355248, 10.84292169980134, 11.59913527206522, 12.03037525663718, 12.71488304375990, 12.83643358176674
