| L(s) = 1 | + 5.26·3-s + 12.1i·7-s + 18.7·9-s − 4.59·11-s + 19.9·13-s − 4.10i·17-s + (17.9 − 6.30i)19-s + 64.0i·21-s − 35.2i·23-s + 51.1·27-s + 16.4i·29-s − 4.01i·31-s − 24.2·33-s − 10.6·37-s + 104.·39-s + ⋯ |
| L(s) = 1 | + 1.75·3-s + 1.73i·7-s + 2.08·9-s − 0.417·11-s + 1.53·13-s − 0.241i·17-s + (0.943 − 0.331i)19-s + 3.05i·21-s − 1.53i·23-s + 1.89·27-s + 0.568i·29-s − 0.129i·31-s − 0.733·33-s − 0.286·37-s + 2.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.460803914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.460803914\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-17.9 + 6.30i)T \) |
| good | 3 | \( 1 - 5.26T + 9T^{2} \) |
| 7 | \( 1 - 12.1iT - 49T^{2} \) |
| 11 | \( 1 + 4.59T + 121T^{2} \) |
| 13 | \( 1 - 19.9T + 169T^{2} \) |
| 17 | \( 1 + 4.10iT - 289T^{2} \) |
| 23 | \( 1 + 35.2iT - 529T^{2} \) |
| 29 | \( 1 - 16.4iT - 841T^{2} \) |
| 31 | \( 1 + 4.01iT - 961T^{2} \) |
| 37 | \( 1 + 10.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 22.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 23.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 97.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 33.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.31T + 4.48e3T^{2} \) |
| 71 | \( 1 - 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7.02iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101.T + 9.40e3T^{2} \) |
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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
−8.899050123939442275013483716069, −8.530775863093271616026102582354, −7.941259750608132622243898028381, −6.88891049969562852145460365142, −5.96736115220108149001014087322, −5.02468750126452596478298405836, −3.90925109267161661621299880663, −2.90932555094951768820247377188, −2.53251646671318797898844686591, −1.37482124823587508823036635625,
1.00475359967031717559886852872, 1.86074467889647470449106263307, 3.37677067507475014080269214826, 3.57720572379146104026582555120, 4.42586668366525214872379761796, 5.76380434633235569724198675210, 6.99595978820069367990119929574, 7.53592775421638215626790431956, 8.070969195457052847599215593733, 8.868065548016183364763601736022
