L(s) = 1 | + i·7-s − 11-s − i·13-s + 19-s + i·23-s − 29-s + 31-s − i·37-s + 41-s + i·43-s + i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯ |
L(s) = 1 | + i·7-s − 11-s − i·13-s + 19-s + i·23-s − 29-s + 31-s − i·37-s + 41-s + i·43-s + i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03370639731 - 0.1186514741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03370639731 - 0.1186514741i\) |
\(L(1)\) |
\(\approx\) |
\(0.8762668408 + 0.07901293174i\) |
\(L(1)\) |
\(\approx\) |
\(0.8762668408 + 0.07901293174i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 \) |
| 67 | \( 1 \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−21.695729722186123722217144840787, −20.70667071978066302046110070260, −20.44193832090846375725059972832, −19.34336765918738300116701301856, −18.62457032467522282732429246898, −17.87728977626029466885205954675, −16.88857896175714051367408175951, −16.37029409830392379318978611046, −15.52373810470186134499015347968, −14.530015090506306356689886754584, −13.7362536131058654810018041606, −13.21449448969374285972205010693, −12.14299395585258436156587219442, −11.30906495796690017115156517039, −10.458257438407715634282286943320, −9.81628750630894019447223195106, −8.790533530090761653504669100433, −7.80084975503973060100116074694, −7.14259328210094764390039955845, −6.24319149951510606003938107342, −5.090933950847478460359713644123, −4.33663347740049202357336110941, −3.349187265836932139192320669087, −2.27595109148774309125493958719, −1.09416969858867942658062225610,
0.02722066220917657539256894249, 1.39107674823859612055520669447, 2.64752570628475380841729316720, 3.19921714888382086478148262472, 4.61310240732977421154928055708, 5.56602332228144094029825940742, 5.96844283045978770557109274956, 7.520981030696059877537707481437, 7.89635926808293584209150858340, 9.05591269585805605146746661873, 9.71717690085353621224893019244, 10.71696828605389383990607568201, 11.49147373120826250900587647961, 12.433477676654488585305465188696, 13.03567887710446278783242529220, 13.93115836022582031282704838198, 14.960519468388983681229495584192, 15.63446686898388133555367539320, 16.09906475611141316912702709014, 17.389898183796880591194302046503, 18.018210113459069614614370346955, 18.63701538799572224074545033708, 19.51334620612594674059909430137, 20.36506234021942362972860644694, 21.15383408466093308156952164623