| L(s) = 1 | + 4.69i·3-s + 7.54·7-s − 13.0·9-s + (5.78 − 9.35i)11-s − 0.829·13-s + 29.8·17-s − 36.2i·19-s + 35.4i·21-s − 29.7i·23-s − 18.9i·27-s + 10.0i·29-s + 34.6·31-s + (43.9 + 27.1i)33-s − 61.9i·37-s − 3.89i·39-s + ⋯ |
| L(s) = 1 | + 1.56i·3-s + 1.07·7-s − 1.44·9-s + (0.525 − 0.850i)11-s − 0.0637·13-s + 1.75·17-s − 1.90i·19-s + 1.68i·21-s − 1.29i·23-s − 0.700i·27-s + 0.346i·29-s + 1.11·31-s + (1.33 + 0.822i)33-s − 1.67i·37-s − 0.0997i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.342939521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342939521\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-5.78 + 9.35i)T \) |
| good | 3 | \( 1 - 4.69iT - 9T^{2} \) |
| 7 | \( 1 - 7.54T + 49T^{2} \) |
| 13 | \( 1 + 0.829T + 169T^{2} \) |
| 17 | \( 1 - 29.8T + 289T^{2} \) |
| 19 | \( 1 + 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 29.7iT - 529T^{2} \) |
| 29 | \( 1 - 10.0iT - 841T^{2} \) |
| 31 | \( 1 - 34.6T + 961T^{2} \) |
| 37 | \( 1 + 61.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 11.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 39.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6.35iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 8.41iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 3.79T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 100.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 66.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 1.65iT - 9.40e3T^{2} \) |
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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
−9.742751728313078323390183695516, −8.954087275229390455807690756914, −8.406403965645817884524761511712, −7.37010976257593535298229262434, −6.09203069853569857812647222221, −5.15718883552806384938603911286, −4.59266667857322881501224142278, −3.65584104114005084935008971426, −2.66373460349110374386882008648, −0.829515958874335624927622064239,
1.36276272872793424305800197059, 1.57393253202348613038079479609, 3.06689935974261009279633362357, 4.39041434988422098757748280470, 5.58325401914092580644596854042, 6.22921966571941195336078055329, 7.38955850320620024258800676602, 7.77677622563457311914279031628, 8.332105731958723532226905172875, 9.634211636541128893106978548597
