L(s) = 1 | + 1.83i·3-s + (−1.86 + 1.23i)5-s − i·7-s − 0.359·9-s + 4.40·11-s + 3.20i·13-s + (−2.26 − 3.41i)15-s − 1.14i·17-s + 1.72·19-s + 1.83·21-s + 3.25i·23-s + (1.94 − 4.60i)25-s + 4.83i·27-s + 4.18·29-s − 1.36·31-s + ⋯ |
L(s) = 1 | + 1.05i·3-s + (−0.833 + 0.553i)5-s − 0.377i·7-s − 0.119·9-s + 1.32·11-s + 0.889i·13-s + (−0.585 − 0.881i)15-s − 0.278i·17-s + 0.395·19-s + 0.399·21-s + 0.678i·23-s + (0.388 − 0.921i)25-s + 0.931i·27-s + 0.777·29-s − 0.245·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.595929850\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.595929850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.86 - 1.23i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 1.83iT - 3T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 - 3.20iT - 13T^{2} \) |
| 17 | \( 1 + 1.14iT - 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 3.25iT - 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 - 4.19iT - 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 2.64iT - 43T^{2} \) |
| 47 | \( 1 + 0.106iT - 47T^{2} \) |
| 53 | \( 1 + 7.86iT - 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.28iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 11.8iT - 73T^{2} \) |
| 79 | \( 1 - 5.48T + 79T^{2} \) |
| 83 | \( 1 - 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−9.433545221854133790067991197975, −8.707476917613690516460861396242, −7.68764662395527225572426400530, −6.96529121965837237044836768027, −6.34299062648313688417475167674, −5.07112719619052551483292670207, −4.14906938161177984449929739177, −3.90917196727909052918837049855, −2.89142493213589924200608534267, −1.27379293387009218385449980060,
0.65380428371892207886212754969, 1.52676065782848001523055422943, 2.80538441568171406587557025781, 3.90255556638583005980735612961, 4.66906806953743767423274970979, 5.83070652431110125939533269536, 6.46423882060505179110477898541, 7.42814515909762260486832961625, 7.80610919129753028431737523730, 8.739978122277900263495364205178