L(s) = 1 | + (1 − 1.73i)3-s + (0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + 2·13-s + 1.99·15-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + 4.00·27-s + 6·29-s + (2 − 3.46i)31-s + (−1 − 1.73i)37-s + (2 − 3.46i)39-s + 6·41-s + ⋯ |
L(s) = 1 | + (0.577 − 0.999i)3-s + (0.223 + 0.387i)5-s + (−0.166 − 0.288i)9-s + 0.554·13-s + 0.516·15-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + 0.769·27-s + 1.11·29-s + (0.359 − 0.622i)31-s + (−0.164 − 0.284i)37-s + (0.320 − 0.554i)39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91288 - 0.948268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91288 - 0.948268i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.924094709902073761501248050588, −8.889299820742350337348374369779, −8.053025319550978287116404368811, −7.45324651381589791605491044968, −6.59858366509367480952496458667, −5.79616924998664611328516655974, −4.55247514025862520893648744209, −3.20179567324733865279262650438, −2.36914283912356409762599381318, −1.11404015255516090492210452155,
1.41411687235421623832430128761, 3.01896128708162683851763325716, 3.82310851333526498030825527579, 4.72867288078326871471190934112, 5.67983218297802034433494315335, 6.65459754637677279776876050756, 7.913492478369081500591229610837, 8.609180564786340551572534074102, 9.308670066528639677666537036848, 10.08148831011459137485763202455