L(s) = 1 | + 1.41i·3-s − 2.82i·5-s − 4·7-s + 0.999·9-s − 1.41i·11-s − 2.82i·13-s + 4.00·15-s − 4·17-s − 7.07i·19-s − 5.65i·21-s − 4·23-s − 3.00·25-s + 5.65i·27-s − 8.48i·29-s + 8·31-s + ⋯ |
L(s) = 1 | + 0.816i·3-s − 1.26i·5-s − 1.51·7-s + 0.333·9-s − 0.426i·11-s − 0.784i·13-s + 1.03·15-s − 0.970·17-s − 1.62i·19-s − 1.23i·21-s − 0.834·23-s − 0.600·25-s + 1.08i·27-s − 1.57i·29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.637339 - 0.637339i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.637339 - 0.637339i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 7.07iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.82iT - 53T^{2} \) |
| 59 | \( 1 + 4.24iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−10.39017818164891154948924488157, −9.739293793711422052090321075488, −9.074550109121390332568244153649, −8.289653010754335617108701445613, −6.89615184515577686861177937745, −5.92858865262993170321113365516, −4.80370649194988237267841040793, −4.04997260810033621865495591668, −2.78191032351616055652616608942, −0.51977093546814629137452395290,
1.92373065670686808966821961592, 3.10550810409748815510461585071, 4.17305747562492982760510545305, 6.06050828383155309270346852197, 6.71202377626927932295426514956, 7.09621972605103215146707564603, 8.275633765158505299243361421439, 9.644120037497726486199151912738, 10.10902537489330026071095475360, 11.06667657864456293587325431395