L(s) = 1 | − 2-s + (−0.403 + 1.68i)3-s + 4-s + (0.403 − 1.68i)6-s + (1.28 + 2.31i)7-s − 8-s + (−2.67 − 1.35i)9-s − 5.34i·11-s + (−0.403 + 1.68i)12-s + 3.95·13-s + (−1.28 − 2.31i)14-s + 16-s − 7.32i·17-s + (2.67 + 1.35i)18-s − 0.807i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.232 + 0.972i)3-s + 0.5·4-s + (0.164 − 0.687i)6-s + (0.484 + 0.874i)7-s − 0.353·8-s + (−0.891 − 0.453i)9-s − 1.61i·11-s + (−0.116 + 0.486i)12-s + 1.09·13-s + (−0.342 − 0.618i)14-s + 0.250·16-s − 1.77i·17-s + (0.630 + 0.320i)18-s − 0.185i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.019420780\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019420780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.403 - 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.28 - 2.31i)T \) |
good | 11 | \( 1 + 5.34iT - 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 7.32iT - 17T^{2} \) |
| 19 | \( 1 + 0.807iT - 19T^{2} \) |
| 23 | \( 1 + 0.281T + 23T^{2} \) |
| 29 | \( 1 + 0.281iT - 29T^{2} \) |
| 31 | \( 1 + 9.07iT - 31T^{2} \) |
| 37 | \( 1 + 6.06iT - 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 - 6.34iT - 43T^{2} \) |
| 47 | \( 1 - 5.78iT - 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 - 6.71iT - 67T^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 + 1.53iT - 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.616492176360658707860310275550, −9.159697760573814132872511748410, −8.479171726878452690840934270149, −7.71586063724285733077729263782, −6.18575896801085250407254245854, −5.79520954437102078695559000203, −4.74533099528161075870711275071, −3.46061631826523788069842822464, −2.57851234807687774706569265391, −0.67065405016812529948573236121,
1.28744276071301248021087525928, 1.92970360979325119435478835175, 3.58474909848827239972003331853, 4.76533131301936567246318851845, 6.02416668395034003510306332384, 6.78333826946622028898908393298, 7.46710889973155753019518365359, 8.211559676290225002783726372949, 8.854248245168340425500098015501, 10.22272487794439469773827572115