| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 12-s + 13-s + 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 23-s + 24-s + 25-s + 26-s + 27-s − 29-s + 30-s + 31-s + 32-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 12-s + 13-s + 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 23-s + 24-s + 25-s + 26-s + 27-s − 29-s + 30-s + 31-s + 32-s + 34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4081 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4081 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.270168564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.270168564\) |
| \(L(1)\) |
\(\approx\) |
\(3.483993407\) |
| \(L(1)\) |
\(\approx\) |
\(3.483993407\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 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
−18.57826782442516663697226517414, −17.87406766818562909960823157691, −16.71667440967585586546726494433, −16.439660964417189238798524526949, −15.389556384687949787084994735927, −14.92367266864026873680991071081, −14.250982277273196680497794908584, −13.64901945442354353508440982692, −13.24036568345006255163791344901, −12.595080557361649075025053348790, −11.78611315954499021427256489881, −10.80740406897256973327244797796, −10.157549076122943538379404562498, −9.61511890863441856309251344220, −8.57218837999489213222022434256, −8.028682411259818106147047767032, −7.14140601671853085521822659548, −6.25727489852639663160216058255, −5.90525429504266016227602015487, −4.84664579042281871956403954668, −4.146119860650611712394368412577, −3.32303895826702083054121560192, −2.750518563286789888458768255047, −1.76471040958674984485974372437, −1.42495298827754090452345046355,
1.42495298827754090452345046355, 1.76471040958674984485974372437, 2.750518563286789888458768255047, 3.32303895826702083054121560192, 4.146119860650611712394368412577, 4.84664579042281871956403954668, 5.90525429504266016227602015487, 6.25727489852639663160216058255, 7.14140601671853085521822659548, 8.028682411259818106147047767032, 8.57218837999489213222022434256, 9.61511890863441856309251344220, 10.157549076122943538379404562498, 10.80740406897256973327244797796, 11.78611315954499021427256489881, 12.595080557361649075025053348790, 13.24036568345006255163791344901, 13.64901945442354353508440982692, 14.250982277273196680497794908584, 14.92367266864026873680991071081, 15.389556384687949787084994735927, 16.439660964417189238798524526949, 16.71667440967585586546726494433, 17.87406766818562909960823157691, 18.57826782442516663697226517414
