L(s) = 1 | − 2-s − i·3-s + 4-s + i·6-s + 2·7-s − 8-s − 9-s − i·12-s + (−2 + 3i)13-s − 2·14-s + 16-s − 2i·17-s + 18-s + 6i·19-s − 2i·21-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 0.755·7-s − 0.353·8-s − 0.333·9-s − 0.288i·12-s + (−0.554 + 0.832i)13-s − 0.534·14-s + 0.250·16-s − 0.485i·17-s + 0.235·18-s + 1.37i·19-s − 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.178312608\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178312608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−9.018004662881399509192646319293, −8.460347589898186358674868817946, −7.79743475197748041815268343052, −6.95939253208227105026924536217, −6.40540195270996919453928869184, −5.27490182008906782889769045452, −4.44317927517669356159017990914, −3.07503171863172281753892584264, −2.03842459472598429319864687694, −1.13070880181262797945631025726,
0.61376638006350763154283655686, 2.11052606940190957352736046954, 3.05789196260715498843694751148, 4.24293063716704901324541428581, 5.11119509728700263407936174600, 5.86414895295749421271660378898, 6.98716070297077277256115575438, 7.69540321604552579539324128677, 8.468212525277996482542995016108, 9.052545989215902396018189445234