| L(s) = 1 | − 0.587·3-s + 0.0220·5-s + 3.20·7-s − 2.65·9-s − 2.26·11-s + 0.668·13-s − 0.0129·15-s − 17-s + 4.04·19-s − 1.88·21-s − 0.862·23-s − 4.99·25-s + 3.32·27-s + 3.45·29-s − 1.93·31-s + 1.33·33-s + 0.0707·35-s − 0.186·37-s − 0.392·39-s + 4.18·41-s − 12.0·43-s − 0.0584·45-s + 1.63·47-s + 3.29·49-s + 0.587·51-s − 2.83·53-s − 0.0499·55-s + ⋯ |
| L(s) = 1 | − 0.339·3-s + 0.00985·5-s + 1.21·7-s − 0.884·9-s − 0.683·11-s + 0.185·13-s − 0.00334·15-s − 0.242·17-s + 0.928·19-s − 0.411·21-s − 0.179·23-s − 0.999·25-s + 0.639·27-s + 0.641·29-s − 0.347·31-s + 0.231·33-s + 0.0119·35-s − 0.0305·37-s − 0.0629·39-s + 0.653·41-s − 1.83·43-s − 0.00871·45-s + 0.238·47-s + 0.471·49-s + 0.0822·51-s − 0.389·53-s − 0.00673·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.587T + 3T^{2} \) |
| 5 | \( 1 - 0.0220T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 - 0.668T + 13T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 + 0.862T + 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 + 1.93T + 31T^{2} \) |
| 37 | \( 1 + 0.186T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 1.63T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 8.22T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 7.29T + 97T^{2} \) |
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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
−7.69156556155728641907767071419, −6.76948272913019431902181898121, −5.97865941977165301906748138813, −5.28100672142042335106258563832, −4.93022406050121589311433291248, −3.96403349363284583551167115758, −3.03716447569858788568912281784, −2.20363270162820358672655191664, −1.26100012301555654582753442500, 0,
1.26100012301555654582753442500, 2.20363270162820358672655191664, 3.03716447569858788568912281784, 3.96403349363284583551167115758, 4.93022406050121589311433291248, 5.28100672142042335106258563832, 5.97865941977165301906748138813, 6.76948272913019431902181898121, 7.69156556155728641907767071419
