| L(s) = 1 | − 5.38·3-s − i·7-s + 19.9·9-s + 14·11-s − 16.1·13-s + 23i·17-s + (−10 + 16.1i)19-s + 5.38i·21-s + i·23-s − 59.2·27-s − 48.4i·29-s + 32.3i·31-s − 75.3·33-s + 32.3·37-s + 86.9·39-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.142i·7-s + 2.22·9-s + 1.27·11-s − 1.24·13-s + 1.35i·17-s + (−0.526 + 0.850i)19-s + 0.256i·21-s + 0.0434i·23-s − 2.19·27-s − 1.67i·29-s + 1.04i·31-s − 2.28·33-s + 0.873·37-s + 2.23·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5179461592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5179461592\) |
| \(L(2)\) |
|
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 | \( 1 + (10 - 16.1i)T \) |
| good | 3 | \( 1 + 5.38T + 9T^{2} \) |
| 7 | \( 1 + iT - 49T^{2} \) |
| 11 | \( 1 - 14T + 121T^{2} \) |
| 13 | \( 1 + 16.1T + 169T^{2} \) |
| 17 | \( 1 - 23iT - 289T^{2} \) |
| 23 | \( 1 - iT - 529T^{2} \) |
| 29 | \( 1 + 48.4iT - 841T^{2} \) |
| 31 | \( 1 - 32.3iT - 961T^{2} \) |
| 37 | \( 1 - 32.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 32.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 26iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.9T + 9.40e3T^{2} \) |
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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
−9.569392850300196247169941825605, −8.509771419876568039368233436905, −7.41560733282847691210935941811, −6.76804374678029755832800460070, −6.01838119390664391761688344969, −5.50418665006538476514017161693, −4.34227618281655049346286685220, −3.96051320713687802834343340740, −2.05178215671201800518529920714, −0.972324984351723281175123201606,
0.22925894315651974448596203637, 1.22485937892824282571502573794, 2.64853105100094016125835197321, 4.18191143335617995355826567517, 4.82387231719546786898293551993, 5.47809215338223679363054140363, 6.47077389979552894102218145931, 6.89914428806131328093114839883, 7.64427390707442793764549417686, 9.118222446205945149946798233398
