| L(s) = 1 | − 4i·7-s + 2·11-s − 4i·13-s + i·17-s + 5·19-s − 5i·23-s − 8·29-s + 7·31-s − 6i·37-s + 6·41-s + 2i·43-s + 8i·47-s − 9·49-s − 9i·53-s − 4·59-s + ⋯ |
| L(s) = 1 | − 1.51i·7-s + 0.603·11-s − 1.10i·13-s + 0.242i·17-s + 1.14·19-s − 1.04i·23-s − 1.48·29-s + 1.25·31-s − 0.986i·37-s + 0.937·41-s + 0.304i·43-s + 1.16i·47-s − 1.28·49-s − 1.23i·53-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.832029897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832029897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 - 17iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.79593909787110178279249813167, −7.34332201287685559925364283471, −6.59068482433198771955878496084, −5.82533640025509501440969514072, −4.97266406245091826938568769950, −4.11691085566333923827286839761, −3.57829159514621225164626331239, −2.63152661745988196379328072350, −1.28528210237523047739542514516, −0.52823090543578266186422143339,
1.31353487338724060151061674662, 2.19025728196716371850517195257, 3.05355969380010303752950436859, 3.93053310083182628517750639645, 4.86075099394627336774032390671, 5.59999648962559276088828868777, 6.13141467073513282250848454377, 6.99970705041457890283100079190, 7.64503171774460451266334485033, 8.570934080550980413168953813353
