L(s) = 1 | + (0.679 + 0.733i)2-s + (−0.279 − 0.960i)3-s + (−0.0771 + 0.997i)4-s + (0.128 + 0.991i)5-s + (0.514 − 0.857i)6-s + (0.978 + 0.204i)7-s + (−0.784 + 0.620i)8-s + (−0.843 + 0.536i)9-s + (−0.640 + 0.767i)10-s + (−0.179 + 0.983i)11-s + (0.978 − 0.204i)12-s + (−0.843 + 0.536i)13-s + (0.514 + 0.857i)14-s + (0.916 − 0.400i)15-s + (−0.988 − 0.153i)16-s + (−0.716 − 0.697i)17-s + ⋯ |
L(s) = 1 | + (0.679 + 0.733i)2-s + (−0.279 − 0.960i)3-s + (−0.0771 + 0.997i)4-s + (0.128 + 0.991i)5-s + (0.514 − 0.857i)6-s + (0.978 + 0.204i)7-s + (−0.784 + 0.620i)8-s + (−0.843 + 0.536i)9-s + (−0.640 + 0.767i)10-s + (−0.179 + 0.983i)11-s + (0.978 − 0.204i)12-s + (−0.843 + 0.536i)13-s + (0.514 + 0.857i)14-s + (0.916 − 0.400i)15-s + (−0.988 − 0.153i)16-s + (−0.716 − 0.697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7237720051 + 1.303645111i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7237720051 + 1.303645111i\) |
\(L(1)\) |
\(\approx\) |
\(1.107135740 + 0.6812870989i\) |
\(L(1)\) |
\(\approx\) |
\(1.107135740 + 0.6812870989i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (0.679 + 0.733i)T \) |
| 3 | \( 1 + (-0.279 - 0.960i)T \) |
| 5 | \( 1 + (0.128 + 0.991i)T \) |
| 7 | \( 1 + (0.978 + 0.204i)T \) |
| 11 | \( 1 + (-0.179 + 0.983i)T \) |
| 13 | \( 1 + (-0.843 + 0.536i)T \) |
| 17 | \( 1 + (-0.716 - 0.697i)T \) |
| 19 | \( 1 + (0.870 - 0.492i)T \) |
| 23 | \( 1 + (-0.988 + 0.153i)T \) |
| 29 | \( 1 + (0.952 - 0.304i)T \) |
| 31 | \( 1 + (-0.0771 + 0.997i)T \) |
| 37 | \( 1 + (-0.558 + 0.829i)T \) |
| 41 | \( 1 + (-0.179 - 0.983i)T \) |
| 43 | \( 1 + (0.870 + 0.492i)T \) |
| 47 | \( 1 + (0.994 + 0.102i)T \) |
| 53 | \( 1 + (0.128 + 0.991i)T \) |
| 59 | \( 1 + (-0.894 - 0.447i)T \) |
| 61 | \( 1 + (0.514 + 0.857i)T \) |
| 67 | \( 1 + (-0.894 + 0.447i)T \) |
| 71 | \( 1 + (-0.0771 - 0.997i)T \) |
| 73 | \( 1 + (0.815 - 0.579i)T \) |
| 79 | \( 1 + (0.916 + 0.400i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.994 + 0.102i)T \) |
| 97 | \( 1 + (-0.988 + 0.153i)T \) |
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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
−24.20383725525336139506500798912, −23.49131396315254468343277054097, −22.267548907928639100796317497543, −21.70081097002333847391543073074, −20.93320811970791317672857266156, −20.27252069102653995486482048902, −19.59661698160767888782482594578, −18.08312698199796757284958297796, −17.26102131005147842061285517655, −16.22117683157588869208375928463, −15.39873158732581452010979261388, −14.378631405067794306954682683284, −13.672136106653356578720291379999, −12.43427693575627987496762258162, −11.69764078902101274910407040333, −10.790324351665924201319071446329, −10.01032125366600072163397356992, −8.97565850669385376525062916869, −8.031761982484806456695054564651, −5.9904172095398611320596313989, −5.29942191405860108806590917245, −4.51276490526894019601552931296, −3.64894801637939866111930308857, −2.21235360401013710047163534448, −0.73364358681409595373864476105,
2.03404178750600784939084843714, 2.77289707291411899756305793045, 4.512380528059808996230434380796, 5.3397661706763280151491569581, 6.53099890996205300271544334636, 7.25603219965203400275319519515, 7.80378257732279188499293967434, 9.14961378178545259233546229633, 10.705102829620800488345722024325, 11.907767252220275486870303266906, 12.12172029770188932637995790822, 13.74539758158869518863335734445, 14.01296120394549996660339192084, 14.993480808709571451544309649247, 15.86569180752743003215981399382, 17.329822778003091258194859302038, 17.78317689212995594433131733026, 18.35276566824948899106307827455, 19.66033205103386633491404097565, 20.71454324970625553013586114882, 21.97051480527685943452610923758, 22.37897878907171448857785870934, 23.3702879540167931921160670398, 24.06571190192237681030222437906, 24.802832376241144837666628897