L(s) = 1 | + (1 + 1.73i)5-s − 2·13-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s − 6·29-s + (4 − 6.92i)31-s + (5 + 8.66i)37-s + 10·41-s + 12·43-s + (−4 − 6.92i)47-s + (3 − 5.19i)53-s + (2 − 3.46i)59-s + (5 + 8.66i)61-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s − 0.554·13-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s − 1.11·29-s + (0.718 − 1.24i)31-s + (0.821 + 1.42i)37-s + 1.56·41-s + 1.82·43-s + (−0.583 − 1.01i)47-s + (0.412 − 0.713i)53-s + (0.260 − 0.450i)59-s + (0.640 + 1.10i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.054709959\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.054709959\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 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 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 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 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.563713325925586368755673513647, −7.60258066342268017206456313319, −7.29173059876157944668133812008, −6.22444304302190242756815919464, −5.76280529475132472950703505109, −4.78831778863037336707564619538, −3.90288006719538502716872330338, −2.81922683282480476876951518483, −2.30345144434421829697701267760, −0.813847327531604889499367665169,
0.912320271006495895192630618682, 1.87780399550877693372696213122, 2.96058864872702659600054619445, 3.99665905324501884866276771816, 4.77122171053755907401490523191, 5.66656881112833724196963872937, 6.02613233175959143762993001913, 7.31504875348357758123373772100, 7.69686573983683101313902698306, 8.678355668756092883105036180071