| L(s) = 1 | + 2.53·3-s + 6.47i·5-s + 1.41·7-s − 2.56·9-s − 4.80i·11-s − 12.8·13-s + 16.4i·15-s − 14.3·17-s + (−16.9 − 8.56i)19-s + 3.60·21-s + 22.4·23-s − 16.9·25-s − 29.3·27-s − 29.2·29-s + 9.22i·31-s + ⋯ |
| L(s) = 1 | + 0.845·3-s + 1.29i·5-s + 0.202·7-s − 0.284·9-s − 0.437i·11-s − 0.986·13-s + 1.09i·15-s − 0.845·17-s + (−0.892 − 0.450i)19-s + 0.171·21-s + 0.975·23-s − 0.677·25-s − 1.08·27-s − 1.00·29-s + 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3066409189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3066409189\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (16.9 + 8.56i)T \) |
| good | 3 | \( 1 - 2.53T + 9T^{2} \) |
| 5 | \( 1 - 6.47iT - 25T^{2} \) |
| 7 | \( 1 - 1.41T + 49T^{2} \) |
| 11 | \( 1 + 4.80iT - 121T^{2} \) |
| 13 | \( 1 + 12.8T + 169T^{2} \) |
| 17 | \( 1 + 14.3T + 289T^{2} \) |
| 23 | \( 1 - 22.4T + 529T^{2} \) |
| 29 | \( 1 + 29.2T + 841T^{2} \) |
| 31 | \( 1 - 9.22iT - 961T^{2} \) |
| 37 | \( 1 + 23.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 7.12iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.54iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 94.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 68.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 51.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 6.63T + 4.48e3T^{2} \) |
| 71 | \( 1 + 44.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 19.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 111. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 136. iT - 9.40e3T^{2} \) |
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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
−9.959680566643277173247187410257, −9.033331701373431910406549561142, −8.482580386343788650808842127692, −7.40811248473912162020998795622, −6.92422193049049218778653715367, −5.93540124178317304042429635916, −4.79430401453306122492257495175, −3.59978405493262329187810426417, −2.79391410867573199736760726223, −2.10863876365302059157820132845,
0.07128863179114387161207579214, 1.67895795993904481965621496484, 2.58148150459641013860958606404, 3.89675072678269361498185193451, 4.75982882769716116819809601300, 5.46312870852319656071671374335, 6.72168095216452138169544574765, 7.72290062961767044715080480432, 8.376917945896295419026883236543, 9.088104053639494808604985442901
