| L(s) = 1 | + 1.41i·2-s − 1.73·3-s − 2.00·4-s − 2.44i·6-s + (3.16 − 6.24i)7-s − 2.82i·8-s + 2.99·9-s − 1.75·11-s + 3.46·12-s + 18.7·13-s + (8.82 + 4.47i)14-s + 4.00·16-s − 23.4·17-s + 4.24i·18-s − 23.0i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.452 − 0.891i)7-s − 0.353i·8-s + 0.333·9-s − 0.159·11-s + 0.288·12-s + 1.44·13-s + (0.630 + 0.319i)14-s + 0.250·16-s − 1.38·17-s + 0.235i·18-s − 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00582 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00582 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7578519508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7578519508\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-3.16 + 6.24i)T \) |
| good | 11 | \( 1 + 1.75T + 121T^{2} \) |
| 13 | \( 1 - 18.7T + 169T^{2} \) |
| 17 | \( 1 + 23.4T + 289T^{2} \) |
| 19 | \( 1 + 23.0iT - 361T^{2} \) |
| 23 | \( 1 - 18.7iT - 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 + 8.60iT - 961T^{2} \) |
| 37 | \( 1 - 70.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 37.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 97.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.43T + 6.88e3T^{2} \) |
| 89 | \( 1 + 42.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 51.7T + 9.40e3T^{2} \) |
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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
−9.389330857801411944092631993432, −8.583049781667069707427940785565, −7.74027394377250749529087910021, −6.84337877421048175294476314135, −6.30066615365886640816874152743, −5.18001233125974338096222386140, −4.44920074153190933096011396680, −3.49545673157748434328507090569, −1.60417878957628496399688387690, −0.26650109519068279188416239077,
1.38421152977048622487233128098, 2.38255923385605402318527166481, 3.74402905928937343598455154314, 4.59567554010414089615932834471, 5.71229561913815764276020377046, 6.20477182964265407471520936867, 7.53349174093131993534307493977, 8.625724438192866710922166612432, 8.971990389889952019940985124189, 10.14691324225751912352947685926
