L(s) = 1 | + (−0.988 − 0.149i)3-s + (−0.365 − 0.930i)5-s + (0.955 + 0.294i)9-s + (−0.955 + 0.294i)11-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (−0.900 + 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.988 − 0.149i)33-s + (0.0747 − 0.997i)37-s + ⋯ |
L(s) = 1 | + (−0.988 − 0.149i)3-s + (−0.365 − 0.930i)5-s + (0.955 + 0.294i)9-s + (−0.955 + 0.294i)11-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (−0.826 − 0.563i)17-s + (−0.5 + 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (−0.900 + 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.988 − 0.149i)33-s + (0.0747 − 0.997i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.005994641967 - 0.2243455038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.005994641967 - 0.2243455038i\) |
\(L(1)\) |
\(\approx\) |
\(0.5098212792 - 0.1613897778i\) |
\(L(1)\) |
\(\approx\) |
\(0.5098212792 - 0.1613897778i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 5 | \( 1 + (-0.365 - 0.930i)T \) |
| 11 | \( 1 + (-0.955 + 0.294i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
| 17 | \( 1 + (-0.826 - 0.563i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.826 + 0.563i)T \) |
| 29 | \( 1 + (-0.900 + 0.433i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.0747 - 0.997i)T \) |
| 41 | \( 1 + (-0.623 + 0.781i)T \) |
| 43 | \( 1 + (-0.623 - 0.781i)T \) |
| 47 | \( 1 + (-0.733 - 0.680i)T \) |
| 53 | \( 1 + (0.0747 + 0.997i)T \) |
| 59 | \( 1 + (0.365 - 0.930i)T \) |
| 61 | \( 1 + (-0.0747 + 0.997i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.733 - 0.680i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (-0.955 - 0.294i)T \) |
| 97 | \( 1 - T \) |
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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
−27.45274059382133107646647017673, −26.44439023530781582820755561179, −25.965492604343645241329821900533, −24.08309926791939555600666922693, −23.768830125035438502623129851008, −22.65229408928702782762095563448, −21.891104426115869447527381042430, −21.11722474823743829403507310236, −19.61812837864569270067926091384, −18.58616768736488448363415746574, −17.99041483527288964605985613326, −16.818344035013922354859529011088, −15.81890388411113985928658371212, −15.08378767923242466026130383742, −13.72795836047823486555840158859, −12.61993963394226464209555427161, −11.34911853213986710496685408023, −10.89742734082139856250800744940, −9.83226621899645772229087361472, −8.29318684986910678507904488597, −6.92884755007289278019490868011, −6.26709695197275186834338014027, −4.867733579166809269181094286044, −3.74293093104868047618454701753, −2.13141587856808806566817080393,
0.18614746575628735811303144647, 1.87062876889792794635345559520, 3.89788261003620480628227180206, 5.08634614723198041959537169409, 5.81643772783084132865647554774, 7.348811610018435165202718393870, 8.23680928658317969466154190686, 9.69675446961219190893091152676, 10.750606695974172207128946236987, 11.76586582084285442043249561400, 12.77308028484442636621790892900, 13.31746126246358168047704647397, 15.20468324272108564979405461530, 15.982826982081798048910541324559, 16.81197780035227245400259911497, 17.847949518972428900445199370300, 18.58189876119825660970833180278, 19.99594920652931208135201687392, 20.74345901203015240172676814210, 21.86716056905524327828509749225, 22.936623357278995639818443723860, 23.59808232755224980120399650737, 24.45545993822531368351865810421, 25.36175012847996017330952731643, 26.76163124173248565748839627600