L(s) = 1 | − i·3-s + 3.46i·5-s + 3.46·7-s − 9-s + 3.46·15-s + 6·17-s − 4i·19-s − 3.46i·21-s − 6.92·23-s − 6.99·25-s + i·27-s + 3.46i·29-s − 3.46·31-s + 11.9i·35-s − 6.92i·37-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.54i·5-s + 1.30·7-s − 0.333·9-s + 0.894·15-s + 1.45·17-s − 0.917i·19-s − 0.755i·21-s − 1.44·23-s − 1.39·25-s + 0.192i·27-s + 0.643i·29-s − 0.622·31-s + 2.02i·35-s − 1.13i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28343 + 0.168967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28343 + 0.168967i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−12.43578162076247118828411620719, −11.43790887345449966666975059859, −10.87565217936545335760641612911, −9.785837093561341675610169907855, −8.186881124766838583071624871498, −7.49608297428456407648782546550, −6.49946332206836102982452761761, −5.24352458317283843766027312129, −3.45352217451139096966025524880, −2.00624925349395138781280071069,
1.53191267870496635938549567217, 3.93684381690113906940585843183, 4.96815686767717604150997629660, 5.73225674221684299987607723230, 7.927093271330810909723914638442, 8.319747243197069991378387465816, 9.519030428496076251758317651144, 10.40632569646133134771252678084, 11.83079820816248540483032063931, 12.16678748678592006463633156510