L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.41 − 1.73i)5-s + (1.74 + 1.98i)7-s − 0.999·8-s + (2.20 + 0.353i)10-s + (4.88 − 2.82i)11-s + (0.902 − 1.56i)13-s + (−0.843 + 2.50i)14-s + (−0.5 − 0.866i)16-s − 4.43i·17-s − 4.20i·19-s + (0.797 + 2.08i)20-s + (4.88 + 2.82i)22-s + (−3.13 + 5.43i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.630 − 0.776i)5-s + (0.661 + 0.750i)7-s − 0.353·8-s + (0.698 + 0.111i)10-s + (1.47 − 0.850i)11-s + (0.250 − 0.433i)13-s + (−0.225 + 0.670i)14-s + (−0.125 − 0.216i)16-s − 1.07i·17-s − 0.964i·19-s + (0.178 + 0.467i)20-s + (1.04 + 0.601i)22-s + (−0.653 + 1.13i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.658329931\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.658329931\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 + 1.73i)T \) |
| 7 | \( 1 + (-1.74 - 1.98i)T \) |
good | 11 | \( 1 + (-4.88 + 2.82i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.902 + 1.56i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.43iT - 17T^{2} \) |
| 19 | \( 1 + 4.20iT - 19T^{2} \) |
| 23 | \( 1 + (3.13 - 5.43i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.728 + 0.420i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.34 + 1.35i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.95iT - 37T^{2} \) |
| 41 | \( 1 + (-5.47 + 9.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (10.1 - 5.88i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.36 + 1.94i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.269T + 53T^{2} \) |
| 59 | \( 1 + (2.74 - 4.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.75 - 3.90i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.51 - 2.02i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.50iT - 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + (-3.09 - 5.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.53 + 3.77i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.72T + 89T^{2} \) |
| 97 | \( 1 + (-7.17 - 12.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.155578023227126874730963233887, −8.537768923838355701937030005618, −7.70004384618370596847678505487, −6.70562293129847042335601954951, −5.83168821710373698727550456232, −5.38686781166811695628380274508, −4.50628032991339503302442228506, −3.50950974265179040103894865093, −2.24453998027837363425152411777, −0.962132682445823420773949045818,
1.48141110601234308786906280876, 1.97616918582027881200210873949, 3.48085896817066308918023027997, 4.10200654693566701631040706525, 4.92812888380052868732943865176, 6.35614480053258944020060288070, 6.43296000201773139902238743877, 7.60670567203821188628486725248, 8.519457558527406146446896213372, 9.488207334738383403427103315863