| L(s) = 1 | + (−1.34 + 3.76i)2-s + (−12.3 − 10.1i)4-s − 11.0·5-s − 11.9i·7-s + (54.8 − 32.9i)8-s + (14.8 − 41.5i)10-s + 219. i·11-s − 37.0·13-s + (44.8 + 16.0i)14-s + (50.2 + 251. i)16-s − 284.·17-s + 45.4i·19-s + (136. + 111. i)20-s + (−826. − 295. i)22-s − 201. i·23-s + ⋯ |
| L(s) = 1 | + (−0.336 + 0.941i)2-s + (−0.773 − 0.633i)4-s − 0.441·5-s − 0.243i·7-s + (0.857 − 0.514i)8-s + (0.148 − 0.415i)10-s + 1.81i·11-s − 0.219·13-s + (0.228 + 0.0818i)14-s + (0.196 + 0.980i)16-s − 0.982·17-s + 0.126i·19-s + (0.341 + 0.279i)20-s + (−1.70 − 0.610i)22-s − 0.380i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6284656973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6284656973\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 - 3.76i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 11.0T + 625T^{2} \) |
| 7 | \( 1 + 11.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 219. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 37.0T + 2.85e4T^{2} \) |
| 17 | \( 1 + 284.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 45.4iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 201. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.22e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.51e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.52e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.63e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 39.9iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.88e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.41e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.83e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.25e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 930. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.16e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.16e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.48e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.11e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.73e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.20e4T + 8.85e7T^{2} \) |
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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
−10.56047683590847604841945707873, −9.821003953780313475322471481826, −8.919195156024677478206720713365, −7.79170941778779410663516039770, −7.17010008945999415028535052769, −6.20966007169071993657753528133, −4.78290833480554970840059961400, −4.15084976233634127350616922830, −2.02679127317376003615178746882, −0.25854737852250913471313861766,
1.01410579823539510921538112376, 2.65025040986514027943101566986, 3.61383727463058642181228719347, 4.81007637457172444559025143345, 6.12899421192198794166706791136, 7.56803142426869745392338596019, 8.578507269955913281856051508191, 9.064714743528084398335757099314, 10.41037833185234626720002944765, 11.07328645146471434561869062036
