| L(s) = 1 | − 0.841i·3-s − 3.29i·7-s + 2.29·9-s + 11-s − i·13-s − 1.15i·17-s − 1.31·19-s − 2.76·21-s − 3.84i·23-s − 4.45i·27-s − 6.61·29-s + 7.58·31-s − 0.841i·33-s + 2.13i·37-s − 0.841·39-s + ⋯ |
| L(s) = 1 | − 0.485i·3-s − 1.24i·7-s + 0.764·9-s + 0.301·11-s − 0.277i·13-s − 0.281i·17-s − 0.302·19-s − 0.604·21-s − 0.800i·23-s − 0.856i·27-s − 1.22·29-s + 1.36·31-s − 0.146i·33-s + 0.350i·37-s − 0.134·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695538697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695538697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.841iT - 3T^{2} \) |
| 7 | \( 1 + 3.29iT - 7T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 1.15iT - 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 + 3.84iT - 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 - 2.13iT - 37T^{2} \) |
| 41 | \( 1 - 6.13T + 41T^{2} \) |
| 43 | \( 1 + 8.81iT - 43T^{2} \) |
| 47 | \( 1 - 2.76iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 - 2.90iT - 67T^{2} \) |
| 71 | \( 1 - 0.866T + 71T^{2} \) |
| 73 | \( 1 + 5.87iT - 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 8.92iT - 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
| 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
−8.745109717454729769172349570268, −7.80418605961448231127341287642, −7.27812224815290513623462610761, −6.65296554520355014180853088230, −5.79674480215247364044781467178, −4.49276643002314290319278834520, −4.10752048657570090405292238511, −2.88370535238925403701773969698, −1.61980154904718765735717831883, −0.61683622340671465032442653125,
1.49978720400721408222974511339, 2.53005724734626292415836899120, 3.65427173296242263932586772139, 4.45402248398471760919136810694, 5.34370269088374668260054891248, 6.08549524826472025108528216971, 6.92096099966049912824611246801, 7.86369149369627104516658922156, 8.644434284970468047269986188724, 9.465067053149660555416499797990
