L(s) = 1 | + (1.09 − 1.34i)3-s + (1.97 − 3.41i)5-s + (0.5 + 0.866i)7-s + (−0.619 − 2.93i)9-s + (0.471 + 0.816i)11-s + (0.5 − 0.866i)13-s + (−2.44 − 6.37i)15-s + 5.60·17-s − 1.28·19-s + (1.71 + 0.272i)21-s + (−2.33 + 4.03i)23-s + (−5.27 − 9.13i)25-s + (−4.62 − 2.36i)27-s + (3.83 + 6.63i)29-s + (3.91 − 6.77i)31-s + ⋯ |
L(s) = 1 | + (0.629 − 0.776i)3-s + (0.881 − 1.52i)5-s + (0.188 + 0.327i)7-s + (−0.206 − 0.978i)9-s + (0.142 + 0.246i)11-s + (0.138 − 0.240i)13-s + (−0.630 − 1.64i)15-s + 1.35·17-s − 0.294·19-s + (0.373 + 0.0593i)21-s + (−0.485 + 0.841i)23-s + (−1.05 − 1.82i)25-s + (−0.890 − 0.455i)27-s + (0.711 + 1.23i)29-s + (0.703 − 1.21i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.372321347\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.372321347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.09 + 1.34i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.97 + 3.41i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.471 - 0.816i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + (2.33 - 4.03i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.83 - 6.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.91 + 6.77i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 + (0.471 - 0.816i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.64 + 4.57i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.22T + 53T^{2} \) |
| 59 | \( 1 + (4.77 - 8.26i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.27 - 9.13i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.858 - 1.48i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 + (8.18 + 14.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.198 + 0.343i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 + (-6.77 - 11.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.547062615109123442452474391245, −8.825853451759660535045540660069, −8.229627958571808060278104374126, −7.42828271202890950332431433949, −6.15395080069466845747982334634, −5.55130311164268437925491274356, −4.55241845560812049500055991791, −3.20617785628403203387857941599, −1.87817635238295622142977836873, −1.11019339614710090052395823283,
1.94736191545826772975331144042, 2.97633204190930929120063450235, 3.69133436713220943153411769472, 4.92657219565155717570637440582, 6.00106561466359806298974499008, 6.76281551019308471092886972685, 7.75135306062867921245961660049, 8.570611767701256758106093194646, 9.639122411310053150133906320886, 10.24071179590703954159427414034