| L(s) = 1 | + i·3-s − i·5-s − 1.23·7-s − 9-s − 6.11·11-s − 0.591·13-s + 15-s + 4.95i·17-s − 7.62·19-s − 1.23i·21-s + (4.72 + 0.821i)23-s − 25-s − i·27-s + 4.66·29-s − 7.50i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.466·7-s − 0.333·9-s − 1.84·11-s − 0.164·13-s + 0.258·15-s + 1.20i·17-s − 1.74·19-s − 0.269i·21-s + (0.985 + 0.171i)23-s − 0.200·25-s − 0.192i·27-s + 0.866·29-s − 1.34i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9887364295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9887364295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-4.72 - 0.821i)T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 13 | \( 1 + 0.591T + 13T^{2} \) |
| 17 | \( 1 - 4.95iT - 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 + 7.50iT - 31T^{2} \) |
| 37 | \( 1 + 2.86iT - 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 5.19iT - 47T^{2} \) |
| 53 | \( 1 - 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 - 2.90T + 67T^{2} \) |
| 71 | \( 1 + 15.7iT - 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 3.91T + 83T^{2} \) |
| 89 | \( 1 - 0.663iT - 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−8.115298447316643334996301554659, −7.70346331206019829115269493312, −6.49939362747176423083787115215, −5.98662862798753312470256325617, −5.11168858756671687006688405370, −4.55808603537254586566679831301, −3.73391791975528323424199160952, −2.77986618610884106937283678161, −2.04489237664429063661329809415, −0.41642468975970863227950842743,
0.60249170608851163724304304745, 2.16152364973792423718580182507, 2.72611116574417390622625975359, 3.41739459989798759743823819253, 4.81103709381609421853961100318, 5.15107204936267465056267750757, 6.22947756732787434611778589574, 6.82424763083343941943989824338, 7.31511371014069997227165803415, 8.239534265626149271106008190118
