L(s) = 1 | − 1.61i·7-s − 9-s − 0.618·11-s + 0.618i·13-s − 1.61·19-s − 0.618i·23-s + 1.61i·37-s − 1.61·41-s + 0.618i·47-s − 1.61·49-s − 1.61i·53-s − 1.61·59-s + 1.61i·63-s + 1.00i·77-s + 81-s + ⋯ |
L(s) = 1 | − 1.61i·7-s − 9-s − 0.618·11-s + 0.618i·13-s − 1.61·19-s − 0.618i·23-s + 1.61i·37-s − 1.61·41-s + 0.618i·47-s − 1.61·49-s − 1.61i·53-s − 1.61·59-s + 1.61i·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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2591473508\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2591473508\) |
\(L(1)\) |
|
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 | \( 1 + 1.61iT - T^{2} \) |
| 11 | \( 1 + 0.618T + T^{2} \) |
| 13 | \( 1 - 0.618iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 + 0.618iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.61iT - T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 0.618iT - T^{2} \) |
| 53 | \( 1 + 1.61iT - T^{2} \) |
| 59 | \( 1 + 1.61T + T^{2} \) |
| 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 | \( 1 + 0.618T + T^{2} \) |
| 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
−8.281800302504254038594602143201, −7.59379287691075400151524391234, −6.62853417019519606750557215045, −6.35781353451169588056136001653, −5.08971323717827158839558591371, −4.48465535260203657476214622750, −3.66849357286705439055417409692, −2.76888523229911173567052149398, −1.63929935785592869824700844551, −0.13130051037305364865967451466,
1.97396195465770005998945690585, 2.63438398402300114061068444108, 3.39993831743869432911254259836, 4.62740817848954176014717518225, 5.57721357462453009047090832336, 5.75608960533784646845657061716, 6.64742423570386658909579677805, 7.75453841086324745601825885864, 8.374087024418579464490554268339, 8.875073035910759696810007381054