| L(s) = 1 | − 2.23i·5-s + 11.8i·7-s − 2.15i·11-s + 21.9·13-s + 18.5i·17-s + 1.30i·19-s + (−22.8 − 2.19i)23-s − 5.00·25-s + 25.1·29-s + 55.8·31-s + 26.6·35-s + 49.6i·37-s + 11.9·41-s − 36.5i·43-s − 10.5·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.69i·7-s − 0.196i·11-s + 1.68·13-s + 1.09i·17-s + 0.0684i·19-s + (−0.995 − 0.0955i)23-s − 0.200·25-s + 0.866·29-s + 1.80·31-s + 0.760·35-s + 1.34i·37-s + 0.290·41-s − 0.849i·43-s − 0.223·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0955 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0955 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.279681287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279681287\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (22.8 + 2.19i)T \) |
| good | 7 | \( 1 - 11.8iT - 49T^{2} \) |
| 11 | \( 1 + 2.15iT - 121T^{2} \) |
| 13 | \( 1 - 21.9T + 169T^{2} \) |
| 17 | \( 1 - 18.5iT - 289T^{2} \) |
| 19 | \( 1 - 1.30iT - 361T^{2} \) |
| 29 | \( 1 - 25.1T + 841T^{2} \) |
| 31 | \( 1 - 55.8T + 961T^{2} \) |
| 37 | \( 1 - 49.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 36.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 73.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 91.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 101. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 21.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 13.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 57.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 35.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 60.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 147. iT - 9.40e3T^{2} \) |
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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
−8.504273633124721148706101170624, −8.170757077118408347352116309150, −6.70273304277459192648992258148, −6.01406937657620435997852747218, −5.72410030145794892561590466169, −4.69537944401905405072561269575, −3.83913768005879788754308416707, −2.91976328894039445780985920415, −1.98518152737682524934908874024, −1.05929747257949494447574534515,
0.55302331767917001362238655203, 1.30227846394927643533332727303, 2.63046480818693384431336124448, 3.59410318357083733963267698067, 4.14743038095556434504021417065, 4.92396095075860564339346948685, 6.18950856847950701078185257470, 6.51988452317429959129628225196, 7.43590292704238353631499840977, 7.88591895783334628667809078830
