| L(s) = 1 | + i·3-s + (−0.890 − 2.05i)5-s + (0.425 − 2.61i)7-s − 9-s − 2.24i·11-s + 2.86·13-s + (2.05 − 0.890i)15-s − 0.466·17-s − 4.40·19-s + (2.61 + 0.425i)21-s + 2.22·23-s + (−3.41 + 3.65i)25-s − i·27-s + 0.668·29-s − 9.53·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.398 − 0.917i)5-s + (0.160 − 0.986i)7-s − 0.333·9-s − 0.677i·11-s + 0.795·13-s + (0.529 − 0.229i)15-s − 0.113·17-s − 1.01·19-s + (0.569 + 0.0928i)21-s + 0.463·23-s + (−0.683 + 0.730i)25-s − 0.192i·27-s + 0.124·29-s − 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8748728350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8748728350\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.890 + 2.05i)T \) |
| 7 | \( 1 + (-0.425 + 2.61i)T \) |
| good | 11 | \( 1 + 2.24iT - 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 + 0.466T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 - 0.668T + 29T^{2} \) |
| 31 | \( 1 + 9.53T + 31T^{2} \) |
| 37 | \( 1 - 6.08iT - 37T^{2} \) |
| 41 | \( 1 + 7.46iT - 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 + 8.72iT - 47T^{2} \) |
| 53 | \( 1 - 3.45iT - 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 + 7.17iT - 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 0.252iT - 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 9.20iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 6.17T + 97T^{2} \) |
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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
−8.871116507301861933652479469052, −8.480982147186117991670174867864, −7.60936819662007417799716010078, −6.64139981541269914431843725450, −5.64081638370203770432076992966, −4.80129237532997785531597699434, −3.98932581906027600391985524572, −3.38637004803209578316801245796, −1.62669154231340719415428762449, −0.32662255883971909958847960560,
1.73271312078997106673933247848, 2.61168815744085281808072658666, 3.60204585639079047291946194120, 4.70740290930084426361510119710, 5.86543265117047828428781547182, 6.43117610490203356374933329458, 7.27731743975266485514402255235, 7.983633619601666419826308106940, 8.801184189602787399531285288649, 9.497249287694423038252631989833
