| L(s) = 1 | − 1.73i·3-s + 2.23·5-s + 2.64i·7-s − 2.99·9-s + 20.0i·11-s + 10.9·13-s − 3.87i·15-s − 31.5·17-s + 0.112i·19-s + 4.58·21-s + 13.6i·23-s + 5.00·25-s + 5.19i·27-s − 57.8·29-s − 59.4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447·5-s + 0.377i·7-s − 0.333·9-s + 1.81i·11-s + 0.839·13-s − 0.258i·15-s − 1.85·17-s + 0.00592i·19-s + 0.218·21-s + 0.595i·23-s + 0.200·25-s + 0.192i·27-s − 1.99·29-s − 1.91i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04622056019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04622056019\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 20.0iT - 121T^{2} \) |
| 13 | \( 1 - 10.9T + 169T^{2} \) |
| 17 | \( 1 + 31.5T + 289T^{2} \) |
| 19 | \( 1 - 0.112iT - 361T^{2} \) |
| 23 | \( 1 - 13.6iT - 529T^{2} \) |
| 29 | \( 1 + 57.8T + 841T^{2} \) |
| 31 | \( 1 + 59.4iT - 961T^{2} \) |
| 37 | \( 1 + 50.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 60.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 21.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 91.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 67.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 0.812T + 5.32e3T^{2} \) |
| 79 | \( 1 + 70.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 128.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 74.2T + 9.40e3T^{2} \) |
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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
−9.051494721451819674203412293647, −7.82048553440556258125008978451, −7.17192297044648381367681968935, −6.41210467274353780601776514606, −5.62198269134721475683695390366, −4.64965322055884774448306250936, −3.70222814328221464365131221111, −2.08574261416335205524790679578, −1.92674717178317288638651733170, −0.01138343387301288226612560233,
1.40502810306070875011733612759, 2.82705877405705979082256404542, 3.65960243870690696965692271836, 4.54424229864624405211908976194, 5.59492193320979487430329489632, 6.22147181463576660284688803675, 7.03159533180016620531317761149, 8.315931946125741360955377363995, 8.803750965097284833060008558611, 9.376906374265582074827296672362
