L(s) = 1 | − i·3-s − 4i·7-s − 9-s + 6i·13-s + 2i·17-s + 4·19-s − 4·21-s − 8i·23-s + i·27-s − 6·29-s − 6i·37-s + 6·39-s + 10·41-s + 4i·43-s − 8i·47-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 1.66i·13-s + 0.485i·17-s + 0.917·19-s − 0.872·21-s − 1.66i·23-s + 0.192i·27-s − 1.11·29-s − 0.986i·37-s + 0.960·39-s + 1.56·41-s + 0.609i·43-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.250703108\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250703108\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 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
−7.67787979128525748295241748992, −7.33302059240082301320787771333, −6.62129982017323761528135039158, −6.05050584953970455265049895544, −4.87009871964939159700663686024, −4.17779799783203680097640235994, −3.54894410971365249500193458135, −2.28847596873125984848867939388, −1.43057612369441307017957108822, −0.35236602085611100523462118348,
1.28667403569307001629132666071, 2.70068670155129956525338989800, 3.03527017017125704532662772327, 4.06509739346937859533138796780, 5.28602581386372295096055403048, 5.47317428905117208141669279883, 6.07343371123732160405667021252, 7.39615762727775282806307568306, 7.84744887332887155401029220398, 8.689624092350373027827152806842