| L(s) = 1 | − 29·3-s + 64·4-s − 61·7-s + 112·9-s + 587·11-s − 1.85e3·12-s + 4.09e3·16-s − 4.20e3·17-s + 1.76e3·21-s + 8.06e3·23-s + 1.56e4·25-s + 1.78e4·27-s − 3.90e3·28-s + 4.67e4·29-s + 4.09e4·31-s − 1.70e4·33-s + 7.16e3·36-s + 1.09e4·37-s + 5.04e3·41-s + 3.75e4·44-s − 1.18e5·48-s − 1.13e5·49-s + 1.21e5·51-s − 1.88e5·59-s + 4.35e5·61-s − 6.83e3·63-s + 2.62e5·64-s + ⋯ |
| L(s) = 1 | − 1.07·3-s + 4-s − 0.177·7-s + 0.153·9-s + 0.441·11-s − 1.07·12-s + 16-s − 0.855·17-s + 0.191·21-s + 0.662·23-s + 25-s + 0.909·27-s − 0.177·28-s + 1.91·29-s + 1.37·31-s − 0.473·33-s + 0.153·36-s + 0.215·37-s + 0.0731·41-s + 0.441·44-s − 1.07·48-s − 0.968·49-s + 0.918·51-s − 0.919·59-s + 1.91·61-s − 0.0273·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.600100672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600100672\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 + p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 3 | \( 1 + 29 T + p^{6} T^{2} \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( 1 + 61 T + p^{6} T^{2} \) |
| 11 | \( 1 - 587 T + p^{6} T^{2} \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 + 4201 T + p^{6} T^{2} \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( 1 - 8066 T + p^{6} T^{2} \) |
| 29 | \( 1 - 46703 T + p^{6} T^{2} \) |
| 31 | \( 1 - 40907 T + p^{6} T^{2} \) |
| 37 | \( 1 - 10919 T + p^{6} T^{2} \) |
| 41 | \( 1 - 5042 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( 1 + 188917 T + p^{6} T^{2} \) |
| 61 | \( 1 - 435287 T + p^{6} T^{2} \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−12.73888726004721102322254094142, −11.81508686804643060040608086287, −11.09921355795549088672496925879, −10.16376435085332043701597616014, −8.507735118152160435828472001179, −6.84277203348738838712420573589, −6.24848908513014973808488801915, −4.81175738156412134916508753519, −2.80801611242587971456981767694, −0.948529302120399260736909302110,
0.948529302120399260736909302110, 2.80801611242587971456981767694, 4.81175738156412134916508753519, 6.24848908513014973808488801915, 6.84277203348738838712420573589, 8.507735118152160435828472001179, 10.16376435085332043701597616014, 11.09921355795549088672496925879, 11.81508686804643060040608086287, 12.73888726004721102322254094142
