L(s) = 1 | + i·3-s + 2·7-s − 9-s − 4i·11-s + 6i·13-s + 6·17-s + 2i·21-s + 4·23-s + 5·25-s − i·27-s + 4i·29-s − 10·31-s + 4·33-s − 2i·37-s − 6·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.755·7-s − 0.333·9-s − 1.20i·11-s + 1.66i·13-s + 1.45·17-s + 0.436i·21-s + 0.834·23-s + 25-s − 0.192i·27-s + 0.742i·29-s − 1.79·31-s + 0.696·33-s − 0.328i·37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56701 + 0.649076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56701 + 0.649076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.71952455842196254021447580282, −9.305349933171789530809623443936, −8.980194246803884730916866604967, −7.937663423525615724717844559466, −6.99709311734244714856378557170, −5.83696971187077453581934174752, −5.03844313933873952201198379022, −4.01758698833620399433183443014, −2.99890171503348173611991854582, −1.36953664321006418648610239834,
1.05358884779538257442246104043, 2.38832576738429158256579444906, 3.61326847361635129245552726008, 5.08102881373563885577068223802, 5.56960999106934779616024988406, 6.98167379803965498680802887920, 7.62636888582029667729632876830, 8.275147052231148820384204512104, 9.368856051707355902230072640025, 10.32746818568192312839748502723