| L(s) = 1 | − i·3-s − 4i·5-s + 7-s − 9-s − 2i·11-s − 6i·13-s − 4·15-s − 4·17-s − 4i·19-s − i·21-s − 2·23-s − 11·25-s + i·27-s − 2i·29-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.78i·5-s + 0.377·7-s − 0.333·9-s − 0.603i·11-s − 1.66i·13-s − 1.03·15-s − 0.970·17-s − 0.917i·19-s − 0.218i·21-s − 0.417·23-s − 2.20·25-s + 0.192i·27-s − 0.371i·29-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375268914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375268914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.81443443276339300719238693457, −7.26484866833276606141366184226, −5.93737545578447518669596129529, −5.71780914359076047139026528655, −4.79050865531242017673957410143, −4.26506201785086260545605186343, −3.05709180159616036834207797884, −2.09098063759233980030122446166, −0.992391089966167412835417857281, −0.40071296808257268154157009603,
1.91694476182441970239571953197, 2.37451747535111190339194820683, 3.53955306345021236489970265512, 4.05908848732914229216413967581, 4.84058802234822855581330556951, 5.92592270945116938395619895548, 6.57842544119891672534374411653, 7.07428814083088902579769386997, 7.76228116116169696420509808612, 8.704198244351956100228088339623
