L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.358 + 2.62i)7-s − 0.999·8-s + (−2.59 − 1.5i)11-s + 2.44·13-s + (−2.44 + i)14-s + (−0.5 − 0.866i)16-s + (−5.12 − 2.95i)17-s + (5.12 − 2.95i)19-s − 3i·22-s + (−2.12 − 3.67i)23-s + (1.22 + 2.12i)26-s + (−2.09 − 1.62i)28-s − 7.24i·29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.135 + 0.990i)7-s − 0.353·8-s + (−0.783 − 0.452i)11-s + 0.679·13-s + (−0.654 + 0.267i)14-s + (−0.125 − 0.216i)16-s + (−1.24 − 0.717i)17-s + (1.17 − 0.678i)19-s − 0.639i·22-s + (−0.442 − 0.766i)23-s + (0.240 + 0.416i)26-s + (−0.395 − 0.306i)28-s − 1.34i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.334347432\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.334347432\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.358 - 2.62i)T \) |
good | 11 | \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + (5.12 + 2.95i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.12 + 3.67i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (7.86 + 4.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.210 - 0.121i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.242iT - 43T^{2} \) |
| 47 | \( 1 + (-5.12 + 2.95i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.62 - 6.27i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.03 - 6.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.878 + 0.507i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.66 - 5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (0.717 - 1.24i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6.63iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.656297277163953354442018419141, −7.76359818949662724520682494547, −7.16602698404231780828339932296, −6.05672843564771857167583922164, −5.80024634683725375385118912512, −4.86626856729106409909985069094, −4.06578387967889399184327774715, −2.91022397039103458350929836249, −2.29927834731904236717894862778, −0.38406659496759285528937503952,
1.17453347937627598133647751560, 2.09527636333670764389006892131, 3.38256929040190902239178832046, 3.84314690421165667684674737207, 4.81275858397330770715900127635, 5.55123821987611986113352098586, 6.44681765626789878969988933490, 7.30621169597926431019558382720, 7.910289335387994406605266677709, 8.912575197267069572813736511039