L(s) = 1 | + (1.30 + 2.25i)2-s + (−2.38 + 4.12i)4-s − 5-s + (−1.80 + 3.11i)7-s − 7.20·8-s + (−1.30 − 2.25i)10-s + (−2.60 − 4.50i)11-s + (3.01 − 1.97i)13-s − 9.37·14-s + (−4.60 − 7.97i)16-s + (−1.46 + 2.54i)17-s + (−3.38 + 5.86i)19-s + (2.38 − 4.12i)20-s + (6.76 − 11.7i)22-s + (2.76 + 4.79i)23-s + ⋯ |
L(s) = 1 | + (0.919 + 1.59i)2-s + (−1.19 + 2.06i)4-s − 0.447·5-s + (−0.680 + 1.17i)7-s − 2.54·8-s + (−0.411 − 0.712i)10-s + (−0.784 − 1.35i)11-s + (0.837 − 0.547i)13-s − 2.50·14-s + (−1.15 − 1.99i)16-s + (−0.356 + 0.616i)17-s + (−0.776 + 1.34i)19-s + (0.533 − 0.923i)20-s + (1.44 − 2.49i)22-s + (0.577 + 0.999i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.554447 - 1.22985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.554447 - 1.22985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (-3.01 + 1.97i)T \) |
good | 2 | \( 1 + (-1.30 - 2.25i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (1.80 - 3.11i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.60 + 4.50i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.46 - 2.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.38 - 5.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.76 - 4.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.916 - 1.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 + (-1.76 - 3.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.68 + 4.65i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.58 - 2.74i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.80T + 47T^{2} \) |
| 53 | \( 1 - 5.20T + 53T^{2} \) |
| 59 | \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.71 + 2.97i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.75 + 3.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.85 - 8.40i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 0.805T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + (-4.91 - 8.51i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.78 + 4.82i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.53955593034454211476727105390, −10.37707238271820198015223404722, −8.829856657203603904689400245153, −8.439964158957402092111372183689, −7.70513940368002865988716992029, −6.32829312566949545356694875164, −5.96506719854977149669227443587, −5.16661243947525911268807148070, −3.75946421614812211971786566235, −3.05953480119591889173353175162,
0.55548320617604482746079748682, 2.23454026947767862374146978917, 3.28515188240909214914391705825, 4.47815315637726637485657618527, 4.68031567596640013103838403058, 6.42362568194526257187568791903, 7.23690890227924756994567334709, 8.767452738946891412862973763136, 9.748509613337287216613076837605, 10.43537252867180251154933297272