L(s) = 1 | + (−0.142 − 0.989i)3-s + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (0.654 + 0.755i)13-s + (−0.415 + 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (0.415 + 0.909i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (0.654 + 0.755i)33-s + (0.959 − 0.281i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)3-s + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (0.654 + 0.755i)13-s + (−0.415 + 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (0.415 + 0.909i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (0.654 + 0.755i)33-s + (0.959 − 0.281i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7598297271 - 0.7424990268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7598297271 - 0.7424990268i\) |
\(L(1)\) |
\(\approx\) |
\(0.8136269076 - 0.1923441082i\) |
\(L(1)\) |
\(\approx\) |
\(0.8136269076 - 0.1923441082i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 + 0.755i)T \) |
| 11 | \( 1 + (-0.841 + 0.540i)T \) |
| 13 | \( 1 + (0.654 + 0.755i)T \) |
| 17 | \( 1 + (-0.415 + 0.909i)T \) |
| 19 | \( 1 + (-0.415 - 0.909i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (0.142 - 0.989i)T \) |
| 37 | \( 1 + (0.959 - 0.281i)T \) |
| 41 | \( 1 + (-0.959 - 0.281i)T \) |
| 43 | \( 1 + (-0.142 - 0.989i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.654 + 0.755i)T \) |
| 61 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (-0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.415 - 0.909i)T \) |
| 79 | \( 1 + (0.654 + 0.755i)T \) |
| 83 | \( 1 + (-0.959 + 0.281i)T \) |
| 89 | \( 1 + (-0.142 - 0.989i)T \) |
| 97 | \( 1 + (0.959 + 0.281i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.40303927356154674103262352900, −23.22970419404416917196161410517, −22.18739647197376434199576879705, −21.38608315820986680173223293254, −20.43340373816436003403578813943, −20.023373748683815795501392211235, −18.73649998497422762994800453868, −17.84811194202568326165717030595, −16.7716978791989036956849974312, −16.10480264282677095974678104822, −15.5460693895666426317387258703, −14.35974193287108627214472115263, −13.53256306326147322424871283859, −12.60976001691156453385705460835, −11.30497301050804897615798975120, −10.53496045006976004110529017874, −9.97322556185051704237956995361, −8.828804077167266458511763204806, −7.93880905575278581950712905089, −6.61231163000596507930158949475, −5.64330502999655198509654599216, −4.671009658616505561102652422581, −3.54401471709577000752707598187, −2.85483416554726988404109653851, −0.807328253470726698493779023230,
0.38124769782622701205943841039, 1.980305201935795849032532037085, 2.64648460761720537266803723355, 4.1578652653622625351710656166, 5.54230952644771907041441290099, 6.32493723556579197235354341250, 7.14635952584702855512651063421, 8.3058505432422109249086452006, 9.016806881335112149941937088360, 10.267550283477744075820871328209, 11.36902807069403553286528194828, 12.15588023171955059219780439813, 13.13543322081278502542029253464, 13.48252742357282823567398120758, 14.92248454244403134194024443550, 15.62200392993535462709332557077, 16.72721726148868766331240297047, 17.62943444225891027627771635985, 18.46710302851130431561383111977, 19.080895607042604766694228475985, 19.852097839133015076439366004757, 20.959834208713311570900084056249, 21.91037597001249738031896235246, 22.76242722637417410361007093125, 23.67827490007080859964785401613