| L(s) = 1 | − i·3-s + (−2.22 − 0.254i)5-s − 2.64i·7-s − 9-s + 1.51·11-s − 3.87i·13-s + (−0.254 + 2.22i)15-s + 3.31i·17-s − 7.08·19-s − 2.64·21-s + 4.82i·23-s + (4.87 + 1.12i)25-s + i·27-s + 2.18·29-s − 7.36·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.993 − 0.113i)5-s − 0.998i·7-s − 0.333·9-s + 0.456·11-s − 1.07i·13-s + (−0.0656 + 0.573i)15-s + 0.803i·17-s − 1.62·19-s − 0.576·21-s + 1.00i·23-s + (0.974 + 0.225i)25-s + 0.192i·27-s + 0.405·29-s − 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2474741207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2474741207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2.22 + 0.254i)T \) |
| good | 7 | \( 1 + 2.64iT - 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + 3.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 7.08T + 19T^{2} \) |
| 23 | \( 1 - 4.82iT - 23T^{2} \) |
| 29 | \( 1 - 2.18T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.87iT - 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 1.01iT - 43T^{2} \) |
| 47 | \( 1 + 7.08iT - 47T^{2} \) |
| 53 | \( 1 - 4.50iT - 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 + 3.60T + 61T^{2} \) |
| 67 | \( 1 - 1.01iT - 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 + 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−8.486482651605003788166626501457, −7.82966316540733773752190945979, −7.30561729174763040827162360661, −6.62263320361609140042863793556, −5.77760822357776606970680230316, −4.81109705073325960855472697876, −3.81208503526375543081378087747, −3.52172316421079925407657100547, −2.08989489194908749608331015571, −0.975444950035737775336272719541,
0.084158929477579871299578668392, 1.89627863966205114191728670980, 2.85576319565174513379196891784, 3.75070158394867103378832900957, 4.54934923302277835294663560257, 5.02653096978341685180516788399, 6.32722940254676118943902337090, 6.63877371824617215329039287842, 7.68736023315401196752913081928, 8.583294844703244982948517127321
