L(s) = 1 | + 3·5-s − 0.302·7-s + 2.69·13-s − 2.30·17-s + 2·19-s + 2.30·23-s + 4·25-s − 7.60·29-s + 6.60·31-s − 0.908·35-s + 0.394·37-s + 6.21·41-s + 9.60·43-s − 1.60·47-s − 6.90·49-s + 4.60·53-s − 7.81·59-s + 6.81·61-s + 8.09·65-s + 8.21·67-s − 3.90·71-s − 3.09·73-s + 6.39·79-s + 1.60·83-s − 6.90·85-s + 13.8·89-s − 0.816·91-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.114·7-s + 0.748·13-s − 0.558·17-s + 0.458·19-s + 0.480·23-s + 0.800·25-s − 1.41·29-s + 1.18·31-s − 0.153·35-s + 0.0648·37-s + 0.970·41-s + 1.46·43-s − 0.234·47-s − 0.986·49-s + 0.632·53-s − 1.01·59-s + 0.872·61-s + 1.00·65-s + 1.00·67-s − 0.463·71-s − 0.361·73-s + 0.719·79-s + 0.176·83-s − 0.749·85-s + 1.46·89-s − 0.0856·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.794652604\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.794652604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.30T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 - 0.394T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 - 4.60T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 61 | \( 1 - 6.81T + 61T^{2} \) |
| 67 | \( 1 - 8.21T + 67T^{2} \) |
| 71 | \( 1 + 3.90T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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
−8.101739136351671933754826703925, −7.31744006795061071169621830091, −6.47718426162445350139061455084, −5.98879049735523798975252584910, −5.36762365329716937398940319490, −4.51976553544408990747927626230, −3.58959185030604405792808054236, −2.64463515503432810562383424323, −1.89342785702541234455822388234, −0.910994184126424918935185357619,
0.910994184126424918935185357619, 1.89342785702541234455822388234, 2.64463515503432810562383424323, 3.58959185030604405792808054236, 4.51976553544408990747927626230, 5.36762365329716937398940319490, 5.98879049735523798975252584910, 6.47718426162445350139061455084, 7.31744006795061071169621830091, 8.101739136351671933754826703925