L(s) = 1 | + (3.45 + 1.99i)3-s + (7.80 − 4.50i)5-s + (5.54 + 4.26i)7-s + (3.44 + 5.96i)9-s + (−8.28 + 14.3i)11-s − 0.446i·13-s + 35.9·15-s + (6.02 + 3.47i)17-s + (−11.0 + 6.37i)19-s + (10.6 + 25.7i)21-s + (−13.2 − 23.0i)23-s + (28.1 − 48.7i)25-s − 8.43i·27-s − 26.4·29-s + (21.7 + 12.5i)31-s + ⋯ |
L(s) = 1 | + (1.15 + 0.664i)3-s + (1.56 − 0.901i)5-s + (0.792 + 0.609i)7-s + (0.382 + 0.662i)9-s + (−0.752 + 1.30i)11-s − 0.0343i·13-s + 2.39·15-s + (0.354 + 0.204i)17-s + (−0.580 + 0.335i)19-s + (0.507 + 1.22i)21-s + (−0.577 − 1.00i)23-s + (1.12 − 1.95i)25-s − 0.312i·27-s − 0.912·29-s + (0.702 + 0.405i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.347764997\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.347764997\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-5.54 - 4.26i)T \) |
good | 3 | \( 1 + (-3.45 - 1.99i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-7.80 + 4.50i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (8.28 - 14.3i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 0.446iT - 169T^{2} \) |
| 17 | \( 1 + (-6.02 - 3.47i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (11.0 - 6.37i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (13.2 + 23.0i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 26.4T + 841T^{2} \) |
| 31 | \( 1 + (-21.7 - 12.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (31.6 + 54.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 0.519iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 25.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-59.4 + 34.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (3.58 - 6.20i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-65.3 - 37.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (39.8 - 23.0i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (21.4 - 37.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 60.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (40.5 + 23.3i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (27.1 + 47.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 11.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-53.1 + 30.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 20.3iT - 9.40e3T^{2} \) |
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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
−10.32097179011549307827892289653, −10.10688167259958235962789635654, −8.947846649480569183396668042873, −8.699107845931986971439887104273, −7.58470728564165706394164095085, −5.97379025605603427968182442326, −5.10057971055895861669135901262, −4.25424136127537056964234502027, −2.44823188149261298188609606393, −1.87311494729622849864744048327,
1.51236667078505371441622674501, 2.49420117276846160082675152648, 3.40208468416246775692783813560, 5.26208325057690775322301469231, 6.20797155310412807629088214176, 7.26080668507333583814637107098, 8.038911883383393519722541092190, 8.898317847131345342923897441330, 9.941324296799489598954872635998, 10.67520243986864244537887950368