L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + i·11-s + 14-s + 16-s + i·17-s − i·19-s + i·22-s − i·23-s + 28-s + 29-s + i·31-s + 32-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + i·11-s + 14-s + 16-s + i·17-s − i·19-s + i·22-s − i·23-s + 28-s + 29-s + i·31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.154272172 + 0.5422164511i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.154272172 + 0.5422164511i\) |
\(L(1)\) |
\(\approx\) |
\(2.289861393 + 0.1425616142i\) |
\(L(1)\) |
\(\approx\) |
\(2.289861393 + 0.1425616142i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 + iT \) |
| 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
−26.80702397246784271226343071250, −25.40961891630136022353309475267, −24.66095937599771657853098812616, −23.84894709901981558547809693122, −23.01405465355358378317281294363, −21.88969845060181793744097262835, −21.15429268123496804920893025819, −20.377897487302434547524306721364, −19.22778338730797082738118550270, −18.07708238554148652674536645283, −16.798026016040682380811650724805, −15.94619830268272538436705109334, −14.804461086087069551907010302415, −14.02734003634959196503183300958, −13.19365120155182791724089662268, −11.75617549227333616075955358397, −11.33099805929893682600447298340, −10.03081356951816795898394259575, −8.36192668101016826764979969718, −7.421201352292703252303805809983, −6.03779913047236765553795641399, −5.13053102890463408887667209519, −3.9680983777510616538695651785, −2.70418847945839487994115409771, −1.26471582840746602121440661680,
1.49905410106539184670477868869, 2.67737895649607083280227612435, 4.29420119251723896708360328797, 4.94527201719432427912956238526, 6.32709602680925752100354254548, 7.38793163811422583963802185430, 8.5214224383076908086469438603, 10.22362373547882841686853741730, 11.130339778626997298292680942826, 12.181221887061126086049300614306, 13.01798189476237298021328167981, 14.26977848491071948051110700304, 14.89598224626864559646170746341, 15.84319143533794741710923797847, 17.1298043461428817687187435971, 17.96960371681773000255637393418, 19.499363102803319496590684781556, 20.32473510127734976196499581321, 21.23717951707375884872727478566, 21.99232598329154129447632868125, 23.12799363256090236474693510741, 23.84374509630856685687239673780, 24.71896451064232958542575874801, 25.61570227991060272504110256821, 26.67367024298937840810585712454