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
−10.70304990246852560922138488376, −9.547859537615544282888871947198, −8.626109573334222940886773693077, −8.211740893937151457620112422164, −7.17083799543508625832577708136, −6.60048927204186084311149702385, −5.83852154358371037655092103998, −4.91288426719213226642582124154, −3.85985158621191175219979699652, −2.03340319150489281258903234508,
0.20704600420972331465068024152, 1.92627424087933218866930952165, 3.01589872454623983117308675566, 3.86874698775196823738142436214, 4.77461277117628869756513046157, 5.71912346303800481733446539975, 7.20947480244239138568305514883, 8.447896473663328187114075263823, 9.100832963575518903740069581076, 9.973831076677426369592596261462