L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 17-s + 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + (0.5 + 0.866i)59-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 17-s + 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s − 53-s + (0.5 + 0.866i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.272017671 + 0.1112871260i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.272017671 + 0.1112871260i\) |
\(L(1)\) |
\(\approx\) |
\(1.081697230 + 0.04672057949i\) |
\(L(1)\) |
\(\approx\) |
\(1.081697230 + 0.04672057949i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−24.682953647242800811582539111856, −23.81660041525145405447921759462, −22.81304355748671558838460175305, −22.36577737586151703628996315908, −21.0177446515040121773611253626, −20.363802075953177070925313684230, −19.40704815574015687919848034906, −18.67768832168453726231453879599, −17.388865003375920644066953822577, −16.80278112310814695023408835739, −15.92118533638729307081994920887, −14.66934166715733217343334809758, −14.05153307139540544618365817525, −12.936153710712416556720076248085, −12.07389377507048020443134647916, −11.082870048939669917559072990709, −9.79059837805952622585469409112, −9.470129373142558976703707392594, −7.81753181876371790964935403916, −7.10602228144460625892270612252, −6.10302331907913606228401963114, −4.68126349924903757870572897062, −3.86055746175332237150053905014, −2.52536027076899948403222837403, −1.05161579647035061975397429932,
1.12278184367559072550863761075, 2.81812588897939852876343766979, 3.504637295458939645710826505152, 5.244096966314844896252860934074, 5.82061698294222593654721951546, 7.09904647778706235634347332046, 8.16923044003555291412085976202, 9.198015674664012019540655992701, 9.97957296132629953677270372087, 11.22882675620198448797196600098, 12.10827284056653299894675330918, 12.927388171852062193279735402587, 14.02868911367276964078811437871, 14.93484609395875500102775368724, 15.87858290080504396820477286390, 16.64894206878174564227354054491, 17.72989181567077389096693710516, 18.659213832352164501368095358965, 19.399773455819232231494311623323, 20.27195139769474409436228848034, 21.47578811789238937275535634303, 22.05039973595120798604880625168, 22.903312295300649472559474682, 23.93142950565993777532961126729, 25.02451366745677149288469436013