| L(s) = 1 | + (−2.95 + 0.5i)3-s − 8i·7-s + (8.5 − 2.95i)9-s + 17.7i·11-s + 2i·13-s − 17.7·17-s − 11·19-s + (4 + 23.6i)21-s − 35.4·23-s + (−23.6 + 13i)27-s − 35.4i·29-s − 46·31-s + (−8.87 − 52.5i)33-s + 16i·37-s + (−1 − 5.91i)39-s + ⋯ |
| L(s) = 1 | + (−0.986 + 0.166i)3-s − 1.14i·7-s + (0.944 − 0.328i)9-s + 1.61i·11-s + 0.153i·13-s − 1.04·17-s − 0.578·19-s + (0.190 + 1.12i)21-s − 1.54·23-s + (−0.876 + 0.481i)27-s − 1.22i·29-s − 1.48·31-s + (−0.268 − 1.59i)33-s + 0.432i·37-s + (−0.0256 − 0.151i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0238742 + 0.160023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0238742 + 0.160023i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.95 - 0.5i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8iT - 49T^{2} \) |
| 11 | \( 1 - 17.7iT - 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 + 17.7T + 289T^{2} \) |
| 19 | \( 1 + 11T + 361T^{2} \) |
| 23 | \( 1 + 35.4T + 529T^{2} \) |
| 29 | \( 1 + 35.4iT - 841T^{2} \) |
| 31 | \( 1 + 46T + 961T^{2} \) |
| 37 | \( 1 - 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 53.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 62iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 35.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16T + 3.72e3T^{2} \) |
| 67 | \( 1 + 113iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 106. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 101iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 68T + 6.24e3T^{2} \) |
| 83 | \( 1 - 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 53.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 22iT - 9.40e3T^{2} \) |
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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
−11.85112931274378496631924742633, −10.98274475009191848179014790726, −10.13395717292301154632788795569, −9.522155116802746718360919871937, −7.84356027685349237923237774835, −6.98575358375508349243388827645, −6.14906916111516207060160817382, −4.60609002384745704697562082634, −4.14065181655108974557940821923, −1.80903108124915669021394032373,
0.085071757099238205817910951762, 2.09128915811338465236388886178, 3.82282415323021516604790355313, 5.39008185217622609256893890710, 5.90526927732681388990142879591, 6.95942774827552603533961551296, 8.372934043276650756535241827613, 9.070924380074935629513078403014, 10.48481631442826818526972236734, 11.13898917366130498728645162184
