| L(s) = 1 | − i·3-s − 3.79i·5-s + 2.15·7-s − 9-s + 2.54i·11-s − 1.95i·13-s − 3.79·15-s + 0.224·17-s + 0.224i·19-s − 2.15i·21-s − 2.82·23-s − 9.42·25-s + i·27-s − 2.62i·29-s − 1.84·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.69i·5-s + 0.816·7-s − 0.333·9-s + 0.766i·11-s − 0.542i·13-s − 0.980·15-s + 0.0545·17-s + 0.0515i·19-s − 0.471i·21-s − 0.589·23-s − 1.88·25-s + 0.192i·27-s − 0.487i·29-s − 0.330·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.308667440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308667440\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.79iT - 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 1.95iT - 13T^{2} \) |
| 17 | \( 1 - 0.224T + 17T^{2} \) |
| 19 | \( 1 - 0.224iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 2.62iT - 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 8.46iT - 61T^{2} \) |
| 67 | \( 1 - 14.7iT - 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 + 5.97T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 1.42T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.342765436381654551342806235905, −7.75229006938247099425341451592, −7.00270339143941557632655921950, −5.81906555822177007500412359363, −5.24348176327349365345844026396, −4.59599161843303560748582141099, −3.73233079738001389071818185934, −2.16080802610381799170493785515, −1.51714152573697640324214817255, −0.39273111951828374290520963555,
1.67913195417614797231818087898, 2.80644089449937185565487791420, 3.42082400333398484311204239755, 4.33693633064703629021155670864, 5.24234261776260177001516539177, 6.22354838497288595376339731032, 6.67011802317632519784179953800, 7.69376601574364777251167385240, 8.185384706701491511704366747146, 9.177730645240211616917209939174
