| L(s) = 1 | − 2.41i·2-s + i·3-s − 3.82·4-s + 2.41·6-s − 0.414i·7-s + 4.41i·8-s − 9-s − 11-s − 3.82i·12-s − 2.82i·13-s − 0.999·14-s + 2.99·16-s − 2.41i·17-s + 2.41i·18-s − 6.41·19-s + ⋯ |
| L(s) = 1 | − 1.70i·2-s + 0.577i·3-s − 1.91·4-s + 0.985·6-s − 0.156i·7-s + 1.56i·8-s − 0.333·9-s − 0.301·11-s − 1.10i·12-s − 0.784i·13-s − 0.267·14-s + 0.749·16-s − 0.585i·17-s + 0.569i·18-s − 1.47·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.232086 + 0.375523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.232086 + 0.375523i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41iT - 2T^{2} \) |
| 7 | \( 1 + 0.414iT - 7T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 2.41iT - 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 0.171iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 11.6iT - 43T^{2} \) |
| 47 | \( 1 - 7.48iT - 47T^{2} \) |
| 53 | \( 1 - 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 0.343iT - 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 + 8.82iT - 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 4.48iT - 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 + 5.82iT - 97T^{2} \) |
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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
−9.973831076677426369592596261462, −9.100832963575518903740069581076, −8.447896473663328187114075263823, −7.20947480244239138568305514883, −5.71912346303800481733446539975, −4.77461277117628869756513046157, −3.86874698775196823738142436214, −3.01589872454623983117308675566, −1.92627424087933218866930952165, −0.20704600420972331465068024152,
2.03340319150489281258903234508, 3.85985158621191175219979699652, 4.91288426719213226642582124154, 5.83852154358371037655092103998, 6.60048927204186084311149702385, 7.17083799543508625832577708136, 8.211740893937151457620112422164, 8.626109573334222940886773693077, 9.547859537615544282888871947198, 10.70304990246852560922138488376
