| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 11-s − 2·14-s + 16-s + 2·17-s − 6·19-s − 22-s − 8·23-s + 2·28-s − 7·31-s − 32-s − 2·34-s − 10·37-s + 6·38-s + 12·41-s + 11·43-s + 44-s + 8·46-s − 12·47-s − 3·49-s + 10·53-s − 2·56-s + 4·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.301·11-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 1.37·19-s − 0.213·22-s − 1.66·23-s + 0.377·28-s − 1.25·31-s − 0.176·32-s − 0.342·34-s − 1.64·37-s + 0.973·38-s + 1.87·41-s + 1.67·43-s + 0.150·44-s + 1.17·46-s − 1.75·47-s − 3/7·49-s + 1.37·53-s − 0.267·56-s + 0.520·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316418696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316418696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−14.45856260259115, −14.13196307045912, −13.29214240775548, −12.69887857185594, −12.30612105124983, −11.72203364113125, −11.26768125224630, −10.61739869115563, −10.44775054315458, −9.645529311896346, −9.220853636607447, −8.621889202614359, −8.118426747335386, −7.754976468132842, −7.119223679265799, −6.502833207159098, −5.948753443474638, −5.401098379721560, −4.701967551097201, −3.892987021052503, −3.636888039573255, −2.396564500440627, −2.087484309865225, −1.366618436570946, −0.4458782924205671,
0.4458782924205671, 1.366618436570946, 2.087484309865225, 2.396564500440627, 3.636888039573255, 3.892987021052503, 4.701967551097201, 5.401098379721560, 5.948753443474638, 6.502833207159098, 7.119223679265799, 7.754976468132842, 8.118426747335386, 8.621889202614359, 9.220853636607447, 9.645529311896346, 10.44775054315458, 10.61739869115563, 11.26768125224630, 11.72203364113125, 12.30612105124983, 12.69887857185594, 13.29214240775548, 14.13196307045912, 14.45856260259115
