| L(s) = 1 | − 2-s − 3·3-s − 4-s + 5-s + 3·6-s + 3·8-s + 6·9-s − 10-s + 3·12-s − 3·15-s − 16-s − 4·17-s − 6·18-s + 2·19-s − 20-s + 9·23-s − 9·24-s + 25-s − 9·27-s − 5·29-s + 3·30-s − 5·32-s + 4·34-s − 6·36-s − 8·37-s − 2·38-s + 3·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s + 0.447·5-s + 1.22·6-s + 1.06·8-s + 2·9-s − 0.316·10-s + 0.866·12-s − 0.774·15-s − 1/4·16-s − 0.970·17-s − 1.41·18-s + 0.458·19-s − 0.223·20-s + 1.87·23-s − 1.83·24-s + 1/5·25-s − 1.73·27-s − 0.928·29-s + 0.547·30-s − 0.883·32-s + 0.685·34-s − 36-s − 1.31·37-s − 0.324·38-s + 0.474·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−15.07689640639921, −14.60694364784926, −13.68627724819892, −13.26441481702885, −13.05982652483636, −12.26095569131990, −11.82711975262560, −11.13306533916628, −10.81734043017994, −10.36548351974537, −9.856014441936949, −9.157316474160542, −8.915296425479940, −8.113408220084659, −7.311923459395921, −6.849098429173888, −6.538031865703480, −5.433299138474298, −5.355763705416527, −4.794768611157680, −4.109128398778251, −3.364751144919079, −2.167289683310842, −1.415001157124036, −0.7665080339975693, 0,
0.7665080339975693, 1.415001157124036, 2.167289683310842, 3.364751144919079, 4.109128398778251, 4.794768611157680, 5.355763705416527, 5.433299138474298, 6.538031865703480, 6.849098429173888, 7.311923459395921, 8.113408220084659, 8.915296425479940, 9.157316474160542, 9.856014441936949, 10.36548351974537, 10.81734043017994, 11.13306533916628, 11.82711975262560, 12.26095569131990, 13.05982652483636, 13.26441481702885, 13.68627724819892, 14.60694364784926, 15.07689640639921
