L(s) = 1 | + 5-s + 7-s + 6.37·11-s + 4.37·13-s + 0.372·17-s + 4.74·19-s − 4.74·23-s + 25-s + 4.37·29-s + 8·31-s + 35-s − 2·37-s − 6.74·41-s + 8.74·43-s − 7.11·47-s + 49-s − 10.7·53-s + 6.37·55-s + 8·59-s − 2.74·61-s + 4.37·65-s + 4·67-s + 8·71-s − 6·73-s + 6.37·77-s − 15.1·79-s − 9.48·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.92·11-s + 1.21·13-s + 0.0902·17-s + 1.08·19-s − 0.989·23-s + 0.200·25-s + 0.811·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s − 1.05·41-s + 1.33·43-s − 1.03·47-s + 0.142·49-s − 1.47·53-s + 0.859·55-s + 1.04·59-s − 0.351·61-s + 0.542·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.726·77-s − 1.70·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.000001631\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.000001631\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.74T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 9.86T + 97T^{2} \) |
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
−8.467059996879404525947152227668, −7.51389081844806909571369654602, −6.53898130747148901259195148314, −6.27445690166024617438760896820, −5.40039871596592190389876787406, −4.43328872388879689377525781247, −3.79290196525935341569016368854, −2.93368673587216142483582339641, −1.61640183045409330966667569580, −1.09891444278881662790486588744,
1.09891444278881662790486588744, 1.61640183045409330966667569580, 2.93368673587216142483582339641, 3.79290196525935341569016368854, 4.43328872388879689377525781247, 5.40039871596592190389876787406, 6.27445690166024617438760896820, 6.53898130747148901259195148314, 7.51389081844806909571369654602, 8.467059996879404525947152227668