L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s + 11-s − 12-s + 2·13-s − 15-s − 16-s + 6·17-s − 18-s + 4·19-s + 20-s − 22-s + 8·23-s + 3·24-s + 25-s − 2·26-s + 27-s + 6·29-s + 30-s − 8·31-s − 5·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 1.66·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 1.43·31-s − 0.883·32-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
\(L(1)\) |
\(\approx\) |
\(1.788919025\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788919025\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.904944012534442715183340028036, −7.42247027190585738466492496691, −6.76776397721290421016504237068, −5.62014618813421886804641668142, −5.02466299668973395006634134988, −4.13010606045164523736628263866, −3.50499361380701044899499541469, −2.76747350953258638863816624792, −1.35625144216134210405011488002, −0.849655894033383385559774997228,
0.849655894033383385559774997228, 1.35625144216134210405011488002, 2.76747350953258638863816624792, 3.50499361380701044899499541469, 4.13010606045164523736628263866, 5.02466299668973395006634134988, 5.62014618813421886804641668142, 6.76776397721290421016504237068, 7.42247027190585738466492496691, 7.904944012534442715183340028036