| L(s) = 1 | + 8·5-s + 24·13-s − 30·17-s + 39·25-s + 40·29-s + 24·37-s + 18·41-s + 49·49-s − 56·53-s + 120·61-s + 192·65-s − 110·73-s − 240·85-s + 78·89-s − 130·97-s + 40·101-s + 120·109-s + 30·113-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 1.84·13-s − 1.76·17-s + 1.55·25-s + 1.37·29-s + 0.648·37-s + 0.439·41-s + 49-s − 1.05·53-s + 1.96·61-s + 2.95·65-s − 1.50·73-s − 2.82·85-s + 0.876·89-s − 1.34·97-s + 0.396·101-s + 1.10·109-s + 0.265·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.221536460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.221536460\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 24 T + p^{2} T^{2} \) |
| 17 | \( 1 + 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 24 T + p^{2} T^{2} \) |
| 41 | \( 1 - 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 120 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 78 T + p^{2} T^{2} \) |
| 97 | \( 1 + 130 T + p^{2} T^{2} \) |
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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
−8.793547564480527934861392934462, −8.405947025598510909935267893161, −7.03375059627796256225375545266, −6.26171262085803484316595172768, −5.97224315078286538302009979467, −4.90890469418208371875497710067, −4.01972625115017998430353695037, −2.79567260269768918616201316775, −1.95562631391396942460901824050, −0.983773415587476290780752467357,
0.983773415587476290780752467357, 1.95562631391396942460901824050, 2.79567260269768918616201316775, 4.01972625115017998430353695037, 4.90890469418208371875497710067, 5.97224315078286538302009979467, 6.26171262085803484316595172768, 7.03375059627796256225375545266, 8.405947025598510909935267893161, 8.793547564480527934861392934462
