| L(s) = 1 | + (−1.93 − 1.11i)5-s + (−3.54 − 6.14i)7-s + (2.92 − 1.68i)11-s + (−1.83 + 3.17i)13-s + 7.23i·17-s − 16.2·19-s + (−19.2 − 11.1i)23-s + (2.5 + 4.33i)25-s + (17.3 − 10.0i)29-s + (−16.5 + 28.6i)31-s + 15.8i·35-s + 19.0·37-s + (−27.9 − 16.1i)41-s + (10.0 + 17.4i)43-s + (−3.71 + 2.14i)47-s + ⋯ |
| L(s) = 1 | + (−0.387 − 0.223i)5-s + (−0.506 − 0.877i)7-s + (0.265 − 0.153i)11-s + (−0.141 + 0.244i)13-s + 0.425i·17-s − 0.852·19-s + (−0.835 − 0.482i)23-s + (0.100 + 0.173i)25-s + (0.599 − 0.346i)29-s + (−0.533 + 0.924i)31-s + 0.453i·35-s + 0.514·37-s + (−0.681 − 0.393i)41-s + (0.234 + 0.406i)43-s + (−0.0790 + 0.0456i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7588575449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7588575449\) |
| \(L(2)\) |
|
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 | \( 1 + (1.93 + 1.11i)T \) |
| good | 7 | \( 1 + (3.54 + 6.14i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.92 + 1.68i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.83 - 3.17i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 7.23iT - 289T^{2} \) |
| 19 | \( 1 + 16.2T + 361T^{2} \) |
| 23 | \( 1 + (19.2 + 11.1i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-17.3 + 10.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (16.5 - 28.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 19.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (27.9 + 16.1i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-10.0 - 17.4i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (3.71 - 2.14i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 41.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-26.3 - 15.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-28.9 - 50.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (39.6 - 68.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 16.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-61.3 - 106. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-83.8 + 48.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-65.3 - 113. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.395932533448613137635526789133, −8.522495417415980731140351095311, −7.891706908801324401984887760258, −6.86218848392584995810322943671, −6.40240011362745425643545192300, −5.22695768799713212895177554540, −4.16304246876220636011814512082, −3.69528778221376456744522993369, −2.35463813171889041272079522032, −0.982129016433755829034507264758,
0.23810381394743847291825042796, 1.95969653982708306794079615480, 2.92390465319088278334959152497, 3.87644168291627673362103910085, 4.87159270265043437042805532957, 5.88851216826442459938346907855, 6.50718708135939539460895318559, 7.47568254888971915576936583084, 8.228213033146132175463937615434, 9.078047164206806507426160300771
