| L(s) = 1 | + 0.732i·5-s − 7-s + 2.73i·11-s + 4i·13-s − 4.19·17-s + 3.46i·19-s + 4.73·23-s + 4.46·25-s − 7.46i·29-s + 2·31-s − 0.732i·35-s + 2i·37-s − 9.66·41-s + 8.92i·43-s − 4·47-s + ⋯ |
| L(s) = 1 | + 0.327i·5-s − 0.377·7-s + 0.823i·11-s + 1.10i·13-s − 1.01·17-s + 0.794i·19-s + 0.986·23-s + 0.892·25-s − 1.38i·29-s + 0.359·31-s − 0.123i·35-s + 0.328i·37-s − 1.50·41-s + 1.36i·43-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7576992106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7576992106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732iT - 5T^{2} \) |
| 11 | \( 1 - 2.73iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 7.46iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 - 8.92iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 8.39iT - 61T^{2} \) |
| 67 | \( 1 - 6.39iT - 67T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 + 8.53T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 2.53iT - 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 2.39T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.740722705491469713642167482031, −8.120741692199124197632969070565, −7.02537845858047408533260225343, −6.76256270997605811286673250319, −6.00127500064347657426610544978, −4.82757637240202361535559488259, −4.35938877990867191647675901647, −3.33550524839960253769420984523, −2.41787389025056310722579970796, −1.49928229187066278733287161660,
0.22024229125814432045093921587, 1.30550245139500752514476159775, 2.77859076710859182529492811548, 3.22786542115732231749849997997, 4.37848092774091786135115726681, 5.18150166404359673909744655740, 5.76356884820427176112488826787, 6.83143372323006382814572684981, 7.15111657924683905772575567439, 8.408907868965730908414939503247
