L(s) = 1 | + i·3-s − 4i·7-s − 9-s − 11-s + 6i·13-s − 6i·17-s + 8·19-s + 4·21-s − i·27-s + 6·29-s − i·33-s − 6i·37-s − 6·39-s − 10·41-s − 8i·43-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 0.301·11-s + 1.66i·13-s − 1.45i·17-s + 1.83·19-s + 0.872·21-s − 0.192i·27-s + 1.11·29-s − 0.174i·33-s − 0.986i·37-s − 0.960·39-s − 1.56·41-s − 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.657791872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.657791872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.65013478411981669343286991700, −7.16790271110017553532003256129, −6.73579524257721472130114487868, −5.59417785446494961163652515086, −4.83360876791575617152765891508, −4.32751423961662575011243128347, −3.54733043450864466714919115250, −2.78840941093918128962596917861, −1.51709663962674736945583071088, −0.46968592589364323727335755822,
1.02953379901027700480141895230, 1.99033277102822886095514260343, 3.05210487621140954362395056101, 3.25660479273900506999309294066, 4.85794366061380761291505487445, 5.37559866717523881095622124380, 5.98904602418638324469940649485, 6.55932841480255888887304010300, 7.63346943361923356497698039418, 8.189895042921327216937712360855