| L(s) = 1 | + (2.49 − 1.66i)3-s + (−0.806 − 4.93i)5-s + (−4.69 − 1.25i)7-s + (3.42 − 8.32i)9-s + (−3.63 − 6.29i)11-s + (4.18 − 1.12i)13-s + (−10.2 − 10.9i)15-s + (−4.69 + 4.69i)17-s + 10.1i·19-s + (−13.8 + 4.70i)21-s + (40.7 − 10.9i)23-s + (−23.7 + 7.95i)25-s + (−5.34 − 26.4i)27-s + (29.2 − 16.8i)29-s + (−11.6 + 20.2i)31-s + ⋯ |
| L(s) = 1 | + (0.830 − 0.556i)3-s + (−0.161 − 0.986i)5-s + (−0.671 − 0.179i)7-s + (0.380 − 0.924i)9-s + (−0.330 − 0.571i)11-s + (0.321 − 0.0861i)13-s + (−0.683 − 0.730i)15-s + (−0.276 + 0.276i)17-s + 0.532i·19-s + (−0.657 + 0.223i)21-s + (1.77 − 0.475i)23-s + (−0.948 + 0.318i)25-s + (−0.198 − 0.980i)27-s + (1.00 − 0.581i)29-s + (−0.376 + 0.652i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0840 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0840 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.14534 - 1.24602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14534 - 1.24602i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.49 + 1.66i)T \) |
| 5 | \( 1 + (0.806 + 4.93i)T \) |
| good | 7 | \( 1 + (4.69 + 1.25i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (3.63 + 6.29i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-4.18 + 1.12i)T + (146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (4.69 - 4.69i)T - 289iT^{2} \) |
| 19 | \( 1 - 10.1iT - 361T^{2} \) |
| 23 | \( 1 + (-40.7 + 10.9i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (-29.2 + 16.8i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (11.6 - 20.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-40.1 + 40.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (-5.28 + 9.15i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (19.2 - 72.0i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (-23.3 - 6.24i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-44.6 - 44.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-11.3 - 6.55i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-27.1 - 46.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-17.0 - 63.7i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 15.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-31.1 - 31.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (124. - 72.0i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (21.4 - 80.0i)T + (-5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 + 124. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-28.4 - 7.61i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−12.64916951256093258909411640902, −11.30149424294486172956097481127, −9.938560400891777544469481802130, −8.899508575730439643714489433644, −8.272580351138848343109495920737, −7.08443456096623366796664327190, −5.85705826510252461221356222820, −4.23687662745383944170748230955, −2.90562880806740557490068427031, −0.983982802648667629071437712962,
2.53732110710680935739017522218, 3.49145034524349813234147556144, 4.93636646549337298689076092288, 6.62940314903747410429678367193, 7.50150072959216616467927649118, 8.795107091607877681227059651817, 9.710792946640494789156191884216, 10.56249437486159326802539628979, 11.49378966484109780654768232950, 12.96347529781829111563378100025
