| L(s) = 1 | − 1.95·3-s + (−2.75 + 4.17i)5-s − 2.18i·7-s − 5.19·9-s − 9.50·11-s + 20.4·13-s + (5.37 − 8.13i)15-s − 6.15i·17-s + (9.68 − 16.3i)19-s + 4.26i·21-s − 10.9i·23-s + (−9.82 − 22.9i)25-s + 27.6·27-s + 28.4i·29-s − 24.8i·31-s + ⋯ |
| L(s) = 1 | − 0.650·3-s + (−0.550 + 0.834i)5-s − 0.312i·7-s − 0.577·9-s − 0.864·11-s + 1.57·13-s + (0.358 − 0.542i)15-s − 0.362i·17-s + (0.509 − 0.860i)19-s + 0.203i·21-s − 0.475i·23-s + (−0.392 − 0.919i)25-s + 1.02·27-s + 0.980i·29-s − 0.800i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.706301 - 0.442062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.706301 - 0.442062i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.75 - 4.17i)T \) |
| 19 | \( 1 + (-9.68 + 16.3i)T \) |
| good | 3 | \( 1 + 1.95T + 9T^{2} \) |
| 7 | \( 1 + 2.18iT - 49T^{2} \) |
| 11 | \( 1 + 9.50T + 121T^{2} \) |
| 13 | \( 1 - 20.4T + 169T^{2} \) |
| 17 | \( 1 + 6.15iT - 289T^{2} \) |
| 23 | \( 1 + 10.9iT - 529T^{2} \) |
| 29 | \( 1 - 28.4iT - 841T^{2} \) |
| 31 | \( 1 + 24.8iT - 961T^{2} \) |
| 37 | \( 1 - 37.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 81.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 70.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 94.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 102. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 73.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 91.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 84.3T + 9.40e3T^{2} \) |
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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
−10.91314369371284399318331843992, −10.56505109526689775073699286703, −9.086267389616193151674073524508, −8.090901125674994865584528816488, −7.11519213651206305496336653277, −6.19516972903136091227618728946, −5.21859041110828848572054193955, −3.83364760082093193492099720497, −2.71266351973069411779947267947, −0.47038436440563266581596509210,
1.18188198788380809185014597188, 3.18204277874404766501403457827, 4.48136620953774394281619555108, 5.59537966528137258290790331633, 6.18680550713455274429637664592, 7.86938470340934053794594758598, 8.357213490020540307063311393855, 9.406657841998503862027279933849, 10.61523682524514853792724153214, 11.41474043077657448767154165152
