| L(s) = 1 | + (−1.62 + 0.586i)3-s + 2.23i·5-s + 2.64·7-s + (2.31 − 1.91i)9-s + 0.359i·11-s − 4.48·13-s + (−1.31 − 3.64i)15-s + 7.99i·17-s + (−4.31 + 1.55i)21-s − 5.00·25-s + (−2.64 + 4.47i)27-s + 10.7i·29-s + (−0.211 − 0.586i)33-s + 5.91i·35-s + (7.31 − 2.63i)39-s + ⋯ |
| L(s) = 1 | + (−0.940 + 0.338i)3-s + 0.999i·5-s + 0.999·7-s + (0.770 − 0.637i)9-s + 0.108i·11-s − 1.24·13-s + (−0.338 − 0.940i)15-s + 1.93i·17-s + (−0.940 + 0.338i)21-s − 1.00·25-s + (−0.509 + 0.860i)27-s + 1.99i·29-s + (−0.0367 − 0.102i)33-s + 0.999i·35-s + (1.17 − 0.421i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.531258 + 0.755862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.531258 + 0.755862i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.62 - 0.586i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 - 0.359iT - 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 - 7.99iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.7iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12.4iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 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
−11.36186430825518251535234910239, −10.50859568825146829537452962402, −10.16360456208305117395380690708, −8.765687200214263594023939302505, −7.55899530893877828887470582456, −6.78745043236790453802561003971, −5.71579523267151009112987626104, −4.77383293807394173205268387165, −3.62700080305182896193176403833, −1.87378073184795797759308463195,
0.67729591294032374535544104254, 2.23295601653530506858218155395, 4.57113516785798083741126238156, 4.94193439259335987969446788672, 5.98210089560113108017988785780, 7.38905208731705342118830072600, 7.88616775502129990580191533486, 9.228337683729837511148001070910, 9.987964664589917021664485546433, 11.25700997649800145010296978329
