| L(s) = 1 | + 5-s − 4·11-s + 6·13-s − 2·17-s + 25-s − 2·29-s − 4·31-s + 10·37-s + 6·41-s − 43-s − 7·49-s − 6·53-s − 4·55-s − 4·59-s + 2·61-s + 6·65-s − 12·67-s − 16·71-s + 14·73-s − 4·79-s − 4·83-s − 2·85-s − 2·89-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 1.64·37-s + 0.937·41-s − 0.152·43-s − 49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.744·65-s − 1.46·67-s − 1.89·71-s + 1.63·73-s − 0.450·79-s − 0.439·83-s − 0.216·85-s − 0.211·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.70274224956121, −13.24502003391533, −12.83077979522672, −12.67046671524342, −11.66396126377983, −11.27266778927184, −10.88857775042733, −10.48700477191360, −9.915408634639194, −9.324101142258558, −8.937244341051607, −8.396308374720724, −7.789904620693448, −7.554318021198248, −6.653881519463882, −6.276769375238911, −5.699427630185372, −5.420028029768082, −4.476143969650308, −4.279174267551584, −3.293301007539183, −3.001651308308388, −2.191408012633152, −1.634245991771821, −0.8942421015721229, 0,
0.8942421015721229, 1.634245991771821, 2.191408012633152, 3.001651308308388, 3.293301007539183, 4.279174267551584, 4.476143969650308, 5.420028029768082, 5.699427630185372, 6.276769375238911, 6.653881519463882, 7.554318021198248, 7.789904620693448, 8.396308374720724, 8.937244341051607, 9.324101142258558, 9.915408634639194, 10.48700477191360, 10.88857775042733, 11.27266778927184, 11.66396126377983, 12.67046671524342, 12.83077979522672, 13.24502003391533, 13.70274224956121
