L(s) = 1 | − 0.488i·2-s + i·3-s + 1.76·4-s + (2.10 − 0.742i)5-s + 0.488·6-s − 5.10i·7-s − 1.83i·8-s − 9-s + (−0.362 − 1.02i)10-s + 1.76i·12-s + 1.81i·13-s − 2.49·14-s + (0.742 + 2.10i)15-s + 2.62·16-s − 0.639i·17-s + 0.488i·18-s + ⋯ |
L(s) = 1 | − 0.345i·2-s + 0.577i·3-s + 0.880·4-s + (0.943 − 0.331i)5-s + 0.199·6-s − 1.93i·7-s − 0.649i·8-s − 0.333·9-s + (−0.114 − 0.325i)10-s + 0.508i·12-s + 0.504i·13-s − 0.666·14-s + (0.191 + 0.544i)15-s + 0.656·16-s − 0.155i·17-s + 0.115i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.550568199\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.550568199\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.10 + 0.742i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.488iT - 2T^{2} \) |
| 7 | \( 1 + 5.10iT - 7T^{2} \) |
| 13 | \( 1 - 1.81iT - 13T^{2} \) |
| 17 | \( 1 + 0.639iT - 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 - 4.37iT - 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 + 5.97iT - 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 3.38iT - 43T^{2} \) |
| 47 | \( 1 - 1.51iT - 47T^{2} \) |
| 53 | \( 1 + 9.17iT - 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 - 7.25iT - 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 + 7.14iT - 73T^{2} \) |
| 79 | \( 1 - 1.90T + 79T^{2} \) |
| 83 | \( 1 - 0.161iT - 83T^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−9.488617649967419481301618918699, −8.348223984127870880450796640145, −7.41704300171178445542242649346, −6.67542109074585360814164888003, −6.05903547846536125785332950876, −4.80636699740372450701844538370, −4.10904092552985032194170290133, −3.16173185204939396061073151790, −1.97055858460148922342844978049, −0.931749352126519444346411150435,
1.63509933833594841144323116760, 2.55512560839777294428392241461, 2.86037546276318756458938236438, 4.91740416800827358466013747466, 5.72101109819777804641765149849, 6.33057400104803714744231766546, 6.65391571854381169404019325034, 7.950634891702658583069525728060, 8.504641332243907498780216675601, 9.214785888575454657867542149159