| L(s) = 1 | + (4 + 6.92i)4-s − 70·13-s + (−31.9 + 55.4i)16-s + (−28 + 48.4i)19-s + (62.5 + 108. i)25-s + (−154 − 266. i)31-s + (−55 + 95.2i)37-s − 520·43-s + (−280 − 484. i)52-s + (−91 + 157. i)61-s − 511.·64-s + (440 + 762. i)67-s + (−595 − 1.03e3i)73-s − 448·76-s + (−442 + 765. i)79-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.866i)4-s − 1.49·13-s + (−0.499 + 0.866i)16-s + (−0.338 + 0.585i)19-s + (0.5 + 0.866i)25-s + (−0.892 − 1.54i)31-s + (−0.244 + 0.423i)37-s − 1.84·43-s + (−0.746 − 1.29i)52-s + (−0.191 + 0.330i)61-s − 0.999·64-s + (0.802 + 1.38i)67-s + (−0.953 − 1.65i)73-s − 0.676·76-s + (−0.629 + 1.09i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7386172822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7386172822\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 70T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (28 - 48.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + (154 + 266. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (55 - 95.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 520T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (91 - 157. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-440 - 762. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + (595 + 1.03e3i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (442 - 765. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−11.30453194257185347936891286881, −10.25081169568286480137101051612, −9.332757193244271912365143230436, −8.251388036442077242208349159031, −7.45779498421284055779773259075, −6.70967973899054809091711450815, −5.41828436421701897590208947581, −4.20949255296742531605535240181, −3.05964471367929791532014632990, −1.94842047405757395210193268107,
0.21407224702063321874449893990, 1.80493315745799743006436770142, 2.92776921553246232766972661783, 4.65829605843546070449210196188, 5.39489699364415509355371136752, 6.64668630759271982649604512118, 7.22313989641582163104713892595, 8.529049672316125847966618328334, 9.567272421880884948901266038398, 10.27956083209803175970728823408
