L(s) = 1 | + (−0.243 − 0.907i)2-s + (−1.70 + 0.315i)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 + (2.80 − 1.07i)9-s + (−0.949 + 1.87i)10-s + (−0.630 + 0.363i)11-s + (−1.47 + 1.25i)12-s + (−1.44 + 1.44i)13-s + (−2.40 + 0.630i)14-s + (3.30 + 2.01i)15-s + (−0.257 + 0.446i)16-s + (7.09 + 1.90i)17-s + ⋯ |
L(s) = 1 | + (−0.171 − 0.641i)2-s + (−0.983 + 0.182i)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 + (0.933 − 0.357i)9-s + (−0.300 + 0.592i)10-s + (−0.189 + 0.109i)11-s + (−0.425 + 0.362i)12-s + (−0.400 + 0.400i)13-s + (−0.642 + 0.168i)14-s + (0.853 + 0.520i)15-s + (−0.0644 + 0.111i)16-s + (1.71 + 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.363555 - 0.585698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.363555 - 0.585698i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.70 - 0.315i)T \) |
| 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
−12.90044929332650853343742724757, −12.16936230863744191001660297733, −11.33171023282774931833584867002, −10.43870275692123547098094365888, −9.604847608018215725663936450362, −7.75198756655275126346824905877, −6.64617777854957867684395599612, −5.12224310180122211295235063380, −3.75056974945045593408872159921, −1.00705138499228596445757631019,
2.96673474953201300034861081821, 5.23311755990776794226069642557, 6.28863793425181451845064929974, 7.36924185026483891304056156992, 8.201307408361321070437559808222, 9.987965749731870337551717183559, 11.26083812192737934319299360360, 11.91145649119444063932010161618, 12.65408435680132099520764479409, 14.53945015683476302046212488934