| L(s) = 1 | + (1.41 + i)3-s − 2.23·5-s + (1.41 + 2.23i)7-s + (1.00 + 2.82i)9-s + 2.82i·11-s − 2.82·13-s + (−3.16 − 2.23i)15-s − 4i·17-s + 6.32i·19-s + (−0.236 + 4.57i)21-s + 6.32·23-s + 5.00·25-s + (−1.41 + 5.00i)27-s + 5.65i·29-s − 6.32i·31-s + ⋯ |
| L(s) = 1 | + (0.816 + 0.577i)3-s − 0.999·5-s + (0.534 + 0.845i)7-s + (0.333 + 0.942i)9-s + 0.852i·11-s − 0.784·13-s + (−0.816 − 0.577i)15-s − 0.970i·17-s + 1.45i·19-s + (−0.0515 + 0.998i)21-s + 1.31·23-s + 1.00·25-s + (−0.272 + 0.962i)27-s + 1.05i·29-s − 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09329 + 1.03834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09329 + 1.03834i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + (-1.41 - 2.23i)T \) |
| good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 6.32iT - 19T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 8.94iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4.47iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4.47iT - 67T^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−11.46017961719289715919198012960, −10.44362511670955091577748824790, −9.456085480020137257301076669029, −8.734488359133466623099554402035, −7.76722872003595646917731268466, −7.17556636581972801956995021586, −5.29225616566483386877377311881, −4.56216033939112187990562861174, −3.37520397796348502742764033762, −2.16988913084909140592314672465,
0.945705598882553938524045934088, 2.80790573159729017515112075252, 3.86181993607126399854839584355, 4.92301287387078587324907338674, 6.72321444534049705017065578242, 7.29338557609884467573699530503, 8.278384835613145705921161287781, 8.766321603375457050949886190117, 10.11436972603748798180135847803, 11.12841566947330159167624353745
