L(s) = 1 | − 0.296·3-s − 5-s + 3.56·7-s − 2.91·9-s − 5.56·11-s + 5.26·13-s + 0.296·15-s + 1.40·17-s − 19-s − 1.05·21-s + 6.96·23-s + 25-s + 1.75·27-s + 1.40·29-s − 1.75·31-s + 1.65·33-s − 3.56·35-s + 3.61·37-s − 1.56·39-s + 4.34·41-s + 3.56·43-s + 2.91·45-s + 8.26·47-s + 5.69·49-s − 0.417·51-s − 7.61·53-s + 5.56·55-s + ⋯ |
L(s) = 1 | − 0.171·3-s − 0.447·5-s + 1.34·7-s − 0.970·9-s − 1.67·11-s + 1.46·13-s + 0.0766·15-s + 0.341·17-s − 0.229·19-s − 0.230·21-s + 1.45·23-s + 0.200·25-s + 0.337·27-s + 0.261·29-s − 0.315·31-s + 0.287·33-s − 0.602·35-s + 0.594·37-s − 0.250·39-s + 0.679·41-s + 0.543·43-s + 0.434·45-s + 1.20·47-s + 0.813·49-s − 0.0584·51-s − 1.04·53-s + 0.750·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.532128382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.532128382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 0.296T + 3T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 - 1.40T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 - 4.15T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.296682033672329674018181599720, −8.332480922053630567729612998642, −8.140745889908473752209333767610, −7.24363724972524303487633719576, −5.99499942851479253306723669856, −5.31219731892161718757037795716, −4.60266806887391510663125215378, −3.37776397168955146406808504636, −2.39338449353553854301783853164, −0.904002215318984343370097964583,
0.904002215318984343370097964583, 2.39338449353553854301783853164, 3.37776397168955146406808504636, 4.60266806887391510663125215378, 5.31219731892161718757037795716, 5.99499942851479253306723669856, 7.24363724972524303487633719576, 8.140745889908473752209333767610, 8.332480922053630567729612998642, 9.296682033672329674018181599720