L(s) = 1 | − 0.618i·7-s − 9-s + 1.61·11-s + 1.61i·13-s + 0.618·19-s − 1.61i·23-s + 0.618i·37-s + 0.618·41-s + 1.61i·47-s + 0.618·49-s − 0.618i·53-s + 0.618·59-s + 0.618i·63-s − 1.00i·77-s + 81-s + ⋯ |
L(s) = 1 | − 0.618i·7-s − 9-s + 1.61·11-s + 1.61i·13-s + 0.618·19-s − 1.61i·23-s + 0.618i·37-s + 0.618·41-s + 1.61i·47-s + 0.618·49-s − 0.618i·53-s + 0.618·59-s + 0.618i·63-s − 1.00i·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.342745298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.342745298\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 - 1.61T + T^{2} \) |
| 13 | \( 1 - 1.61iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 + 1.61iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618iT - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.61iT - T^{2} \) |
| 53 | \( 1 + 0.618iT - T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.801790051693210850385347406114, −7.948321786656307458706766535921, −6.93690430752401324142094483678, −6.55656765581435062934760234396, −5.83882136065378387837589732836, −4.62108847752277971524425784973, −4.16410299623851445200865800315, −3.25454378175731826934697389640, −2.16660993313762805158250279725, −1.04841541239011216516111618840,
1.00443465304627483002935169179, 2.27878941582784016870294192260, 3.27945918539258542339187992909, 3.79257508186621277055952131660, 5.15339221181815923094476856809, 5.65174714330096384006790224702, 6.23723442122574676473765063285, 7.26278168039049891627197832289, 7.907074565090729262639057398264, 8.768706667180122260647628614832