L(s) = 1 | − 2·5-s − 2·7-s + 4·11-s − 2·13-s − 2·17-s + 2·19-s − 23-s − 25-s − 4·29-s + 4·35-s + 2·37-s + 2·43-s + 12·47-s − 3·49-s + 2·53-s − 8·55-s + 12·59-s − 14·61-s + 4·65-s − 2·67-s + 6·73-s − 8·77-s − 6·79-s + 4·83-s + 4·85-s + 18·89-s + 4·91-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.208·23-s − 1/5·25-s − 0.742·29-s + 0.676·35-s + 0.328·37-s + 0.304·43-s + 1.75·47-s − 3/7·49-s + 0.274·53-s − 1.07·55-s + 1.56·59-s − 1.79·61-s + 0.496·65-s − 0.244·67-s + 0.702·73-s − 0.911·77-s − 0.675·79-s + 0.439·83-s + 0.433·85-s + 1.90·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.183624283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183624283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.756820388022993706386597883689, −7.72329865353503515255035549943, −7.23277609268390720855549873758, −6.44895923236525547832891059823, −5.72471111599327933893070064294, −4.58586433816909266989385359452, −3.90905196602135692906010200843, −3.24093317660364111324335059880, −2.06822255512359606964443866509, −0.63643084098289266925839327047,
0.63643084098289266925839327047, 2.06822255512359606964443866509, 3.24093317660364111324335059880, 3.90905196602135692906010200843, 4.58586433816909266989385359452, 5.72471111599327933893070064294, 6.44895923236525547832891059823, 7.23277609268390720855549873758, 7.72329865353503515255035549943, 8.756820388022993706386597883689