L(s) = 1 | − i·2-s + 1.30i·3-s − 4-s + 1.30·6-s − 0.302i·7-s + i·8-s + 1.30·9-s + 4.60·11-s − 1.30i·12-s − 1.30i·13-s − 0.302·14-s + 16-s − 2.30i·17-s − 1.30i·18-s − 6.60·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.752i·3-s − 0.5·4-s + 0.531·6-s − 0.114i·7-s + 0.353i·8-s + 0.434·9-s + 1.38·11-s − 0.376i·12-s − 0.361i·13-s − 0.0809·14-s + 0.250·16-s − 0.558i·17-s − 0.307i·18-s − 1.51·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.784016725\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784016725\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 1.30iT - 3T^{2} \) |
| 7 | \( 1 + 0.302iT - 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 + 1.30iT - 13T^{2} \) |
| 17 | \( 1 + 2.30iT - 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 + 3.90iT - 23T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.60iT - 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 0.302iT - 43T^{2} \) |
| 47 | \( 1 + 9.21iT - 47T^{2} \) |
| 53 | \( 1 - 9.69iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 3.30T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 1.90T + 79T^{2} \) |
| 83 | \( 1 + 7.81iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 9.11iT - 97T^{2} \) |
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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
−9.573951191825616422015654732021, −8.896228826786564593252610850945, −8.174594712650930415695353148814, −6.89600837710128038279716916555, −6.21772236497891081578530631090, −4.84119530760227908915465267476, −4.31935601468584348403480455592, −3.52855398452850451515124902677, −2.33639587063362683445267341666, −0.968309749179320741965891951673,
1.07662233890556900058323718633, 2.19773890167835570569176934007, 3.88008541843714420732471338040, 4.42803433230484454317659436846, 5.77685282411163153341699446803, 6.56772075218612425649020988405, 6.89758752111149639965100540499, 7.952801899701666391133925083357, 8.585120469202120106853607938620, 9.407505941436812951230021169120