L(s) = 1 | + (−2.68 − 2.68i)5-s − 2.15i·7-s + (−1.79 − 1.79i)11-s + (−1.38 + 1.38i)13-s + 0.224·17-s + (0.158 − 0.158i)19-s + 2.82i·23-s + 9.42i·25-s + (−1.85 + 1.85i)29-s − 1.84·31-s + (−5.79 + 5.79i)35-s + (3.66 + 3.66i)37-s + 5.88i·41-s + (−7.75 − 7.75i)43-s − 2.82·47-s + ⋯ |
L(s) = 1 | + (−1.20 − 1.20i)5-s − 0.816i·7-s + (−0.542 − 0.542i)11-s + (−0.383 + 0.383i)13-s + 0.0545·17-s + (0.0364 − 0.0364i)19-s + 0.589i·23-s + 1.88i·25-s + (−0.344 + 0.344i)29-s − 0.330·31-s + (−0.980 + 0.980i)35-s + (0.603 + 0.603i)37-s + 0.918i·41-s + (−1.18 − 1.18i)43-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1589065108\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1589065108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.68 + 2.68i)T + 5iT^{2} \) |
| 7 | \( 1 + 2.15iT - 7T^{2} \) |
| 11 | \( 1 + (1.79 + 1.79i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.38 - 1.38i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.224T + 17T^{2} \) |
| 19 | \( 1 + (-0.158 + 0.158i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (1.85 - 1.85i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + (-3.66 - 3.66i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.88iT - 41T^{2} \) |
| 43 | \( 1 + (7.75 + 7.75i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-7.51 - 7.51i)T + 53iT^{2} \) |
| 59 | \( 1 + (4 + 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.98 - 5.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (10.4 - 10.4i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.31iT - 71T^{2} \) |
| 73 | \( 1 + 5.97iT - 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.42iT - 89T^{2} \) |
| 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
−9.126359101408692396621965726193, −8.440003844067960334781880224639, −7.67089171703012738716026983740, −7.11037201055097159930185319386, −5.73333367201342376509700211969, −4.78919801505605139650791820307, −4.12931082824143592239418979029, −3.17334972197940613648439364282, −1.34075838627652567919648085986, −0.07195221270101893830270542546,
2.30754610283712418172933294009, 3.09345277628479565369440587227, 4.11522475340540211328779965761, 5.17231059678314682889949049171, 6.22570094906515276126576676141, 7.13977311407295097565354214770, 7.75036516426314019807721227092, 8.477068744407288761114101878421, 9.576134203590517127369459664789, 10.41382152462959261994615092360