| L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 11-s − 4·13-s − 4·15-s + 17-s + 19-s + 3·23-s − 25-s + 4·27-s − 2·29-s + 4·31-s + 2·33-s − 4·37-s + 8·39-s − 2·41-s + 4·43-s + 2·45-s + 47-s − 2·51-s + 2·53-s − 2·55-s − 2·57-s − 4·59-s + 15·61-s − 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 1.03·15-s + 0.242·17-s + 0.229·19-s + 0.625·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s + 0.348·33-s − 0.657·37-s + 1.28·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 0.145·47-s − 0.280·51-s + 0.274·53-s − 0.269·55-s − 0.264·57-s − 0.520·59-s + 1.92·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.33430604262879797635980709066, −6.75129168683268613980373677115, −6.04663441205070151082737143336, −5.35271233379837671800453953599, −5.10555842138756039152515277292, −4.15133095045256086838417496303, −2.96491946975427332357068570569, −2.22845374613087836253609245940, −1.13471701029530725484701395747, 0,
1.13471701029530725484701395747, 2.22845374613087836253609245940, 2.96491946975427332357068570569, 4.15133095045256086838417496303, 5.10555842138756039152515277292, 5.35271233379837671800453953599, 6.04663441205070151082737143336, 6.75129168683268613980373677115, 7.33430604262879797635980709066
