L(s) = 1 | − 5.19·3-s − 8i·4-s + (−25.5 − 25.5i)7-s + 27·9-s + 41.5i·12-s + (31.1 + 35i)13-s − 64·16-s + (49.9 − 49.9i)19-s + (132. + 132. i)21-s − 125i·25-s − 140.·27-s + (−204. + 204. i)28-s + (231. − 231. i)31-s − 216i·36-s + (−163. − 163. i)37-s + ⋯ |
L(s) = 1 | − 1.00·3-s − i·4-s + (−1.38 − 1.38i)7-s + 9-s + 0.999i·12-s + (0.665 + 0.746i)13-s − 16-s + (0.603 − 0.603i)19-s + (1.38 + 1.38i)21-s − i·25-s − 1.00·27-s + (−1.38 + 1.38i)28-s + (1.34 − 1.34i)31-s − i·36-s + (−0.725 − 0.725i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.521 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.336299 - 0.600030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.336299 - 0.600030i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 13 | \( 1 + (-31.1 - 35i)T \) |
good | 2 | \( 1 + 8iT^{2} \) |
| 5 | \( 1 + 125iT^{2} \) |
| 7 | \( 1 + (25.5 + 25.5i)T + 343iT^{2} \) |
| 11 | \( 1 - 1.33e3iT^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + (-49.9 + 49.9i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + (-231. + 231. i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (163. + 163. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 6.89e4iT^{2} \) |
| 43 | \( 1 - 218. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 1.03e5iT^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 - 2.05e5iT^{2} \) |
| 61 | \( 1 - 935.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-112. + 112. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 3.57e5iT^{2} \) |
| 73 | \( 1 + (407. + 407. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5iT^{2} \) |
| 89 | \( 1 - 7.04e5iT^{2} \) |
| 97 | \( 1 + (-20.8 + 20.8i)T - 9.12e5iT^{2} \) |
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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
−15.74919646684040088193110665381, −13.95877025949219821621648341835, −13.11492955502056318331339053989, −11.47602965085814111631916940223, −10.38085660668039823711397929570, −9.606147852636555739952188230441, −6.96658573105494420842171104876, −6.12679362412166326822274238597, −4.29787692687587276082604923756, −0.66419974701740016513030352077,
3.28808317311945961300263030970, 5.55138954005615592418663929417, 6.82179053380510227777450328860, 8.601769481558970937853468140015, 10.01009997448948708431825629532, 11.67247590116403927743192449948, 12.45585653964159330141840146826, 13.25731912265037693548921827382, 15.61816686332591610876954141705, 16.01574432766853867475195726766