L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 6·11-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 4·19-s + 20-s − 6·22-s + 25-s + 4·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 2·35-s + 8·37-s + 4·38-s − 40-s + 8·43-s + 6·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.338·35-s + 1.31·37-s + 0.648·38-s − 0.158·40-s + 1.21·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8199784570\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8199784570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.47336871923298780011111515198, −12.97148624749448870913856837974, −11.71789183657864035735784262261, −10.91484441178597888410818702330, −9.508626637108993943325064267957, −8.808531037524113106929321069025, −7.33859557219653016096826874551, −6.21060412222094990512561243657, −4.39745787492882284514235740312, −1.98468742030680472449906296868,
1.98468742030680472449906296868, 4.39745787492882284514235740312, 6.21060412222094990512561243657, 7.33859557219653016096826874551, 8.808531037524113106929321069025, 9.508626637108993943325064267957, 10.91484441178597888410818702330, 11.71789183657864035735784262261, 12.97148624749448870913856837974, 14.47336871923298780011111515198