L(s) = 1 | + 2.23i·2-s − 3.00·4-s + 2.23i·5-s − 2.23i·8-s − 5.00·10-s − 0.999·16-s + 4.47i·17-s + 4·19-s − 6.70i·20-s + 8.94i·23-s − 5.00·25-s − 8·31-s − 6.70i·32-s − 10.0·34-s + 8.94i·38-s + ⋯ |
L(s) = 1 | + 1.58i·2-s − 1.50·4-s + 0.999i·5-s − 0.790i·8-s − 1.58·10-s − 0.249·16-s + 1.08i·17-s + 0.917·19-s − 1.50i·20-s + 1.86i·23-s − 1.00·25-s − 1.43·31-s − 1.18i·32-s − 1.71·34-s + 1.45i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.003115048\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003115048\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8.94iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.459501872335897529754981238442, −8.600807646800967871459093888562, −7.77182765048689566584365085880, −7.27481775856082839766551623409, −6.66161758916153036436930111346, −5.71907810840089063330202393637, −5.39105757960779777521960309001, −4.04715576888922035705059558833, −3.29610816047882424687748144418, −1.84868146419156580856953688642,
0.35875281877674739545711697242, 1.34085808902525001930383209282, 2.41924016286910073489575241045, 3.28581702853468131659674010390, 4.34238186933368609338011079235, 4.85524717981132801762920878720, 5.81923015699735430038580931572, 7.03577164825596187182892335619, 7.949009327963947257185077913757, 9.007523399322196319546071085346