| L(s) = 1 | − 3-s − 2·4-s + 5-s + 9-s − 11-s + 2·12-s + 4·13-s − 15-s + 4·16-s − 3·17-s + 19-s − 2·20-s − 3·23-s + 25-s − 27-s − 9·29-s + 10·31-s + 33-s − 2·36-s − 4·37-s − 4·39-s − 6·41-s − 43-s + 2·44-s + 45-s + 6·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 1.10·13-s − 0.258·15-s + 16-s − 0.727·17-s + 0.229·19-s − 0.447·20-s − 0.625·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s + 1.79·31-s + 0.174·33-s − 1/3·36-s − 0.657·37-s − 0.640·39-s − 0.937·41-s − 0.152·43-s + 0.301·44-s + 0.149·45-s + 0.875·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.55991498762763144115229862899, −6.55769751461115499659729065536, −6.06797360529370022184298478534, −5.35192752470755626537426653772, −4.76840681188539894008591283034, −3.98385591832454545209055086978, −3.31108706122565202823011397372, −2.06259595740741369885115115509, −1.10476373833240213674711830048, 0,
1.10476373833240213674711830048, 2.06259595740741369885115115509, 3.31108706122565202823011397372, 3.98385591832454545209055086978, 4.76840681188539894008591283034, 5.35192752470755626537426653772, 6.06797360529370022184298478534, 6.55769751461115499659729065536, 7.55991498762763144115229862899
