L(s) = 1 | + (0.243 + 0.907i)2-s + (−1.31 − 1.12i)3-s + (0.967 − 0.558i)4-s + (1.66 + 1.49i)5-s + (0.700 − 1.46i)6-s + (0.0144 − 2.64i)7-s + (2.07 + 2.07i)8-s + (0.470 + 2.96i)9-s + (−0.949 + 1.87i)10-s + (0.630 − 0.363i)11-s + (−1.90 − 0.352i)12-s + (−1.44 + 1.44i)13-s + (2.40 − 0.630i)14-s + (−0.515 − 3.83i)15-s + (−0.257 + 0.446i)16-s + (−7.09 − 1.90i)17-s + ⋯ |
L(s) = 1 | + (0.171 + 0.641i)2-s + (−0.760 − 0.649i)3-s + (0.483 − 0.279i)4-s + (0.744 + 0.667i)5-s + (0.285 − 0.599i)6-s + (0.00544 − 0.999i)7-s + (0.732 + 0.732i)8-s + (0.156 + 0.987i)9-s + (−0.300 + 0.592i)10-s + (0.189 − 0.109i)11-s + (−0.549 − 0.101i)12-s + (−0.400 + 0.400i)13-s + (0.642 − 0.168i)14-s + (−0.133 − 0.991i)15-s + (−0.0644 + 0.111i)16-s + (−1.71 − 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09887 + 0.122113i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09887 + 0.122113i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.31 + 1.12i)T \) |
| 5 | \( 1 + (-1.66 - 1.49i)T \) |
| 7 | \( 1 + (-0.0144 + 2.64i)T \) |
good | 2 | \( 1 + (-0.243 - 0.907i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-0.630 + 0.363i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.44 - 1.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (7.09 + 1.90i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.664 + 0.383i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.13 - 0.840i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.08 + 1.63i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.44iT - 41T^{2} \) |
| 43 | \( 1 + (5.15 - 5.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.82 - 6.79i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.41 + 5.26i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.807 + 1.39i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.84 + 6.90i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.06iT - 71T^{2} \) |
| 73 | \( 1 + (-15.2 - 4.08i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.80 + 3.35i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.83 + 1.83i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \) |
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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
−13.82644195498246144195401129205, −13.10740430401115278599030400520, −11.36585604368992257682223897194, −10.94645698602392736905813221196, −9.750815328162047964493103269303, −7.71457861236073919943853818382, −6.78560622612224371978551636261, −6.22553101429231411132852544659, −4.77478235978827560227104249067, −2.04814462909678929295636779610,
2.22442289446052811584306463215, 4.19793706280072383652498011781, 5.53383447784735967982902130991, 6.63830762211975363097745162227, 8.620144448583061485018753819444, 9.694009145111298554574264829329, 10.68305007261624622088859968040, 11.71759249037398741037596282571, 12.46732676408580339138999508366, 13.27953235084798891518565764900