| L(s) = 1 | + 7-s + 6·11-s + 13-s − 19-s − 6·23-s + 6·29-s + 8·31-s + 7·37-s − 6·41-s + 4·43-s − 12·47-s − 6·49-s + 6·53-s + 11·61-s + 7·67-s − 6·71-s − 11·73-s + 6·77-s − 79-s − 6·83-s − 12·89-s + 91-s + 13·97-s + 13·103-s + 18·107-s − 10·109-s + 18·113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.80·11-s + 0.277·13-s − 0.229·19-s − 1.25·23-s + 1.11·29-s + 1.43·31-s + 1.15·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s − 6/7·49-s + 0.824·53-s + 1.40·61-s + 0.855·67-s − 0.712·71-s − 1.28·73-s + 0.683·77-s − 0.112·79-s − 0.658·83-s − 1.27·89-s + 0.104·91-s + 1.31·97-s + 1.28·103-s + 1.74·107-s − 0.957·109-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.198335067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.198335067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.531058659433827241315008209209, −8.413818795997672639773834461783, −7.25266822370393144487599476460, −6.42497239306053748570404845338, −6.01234397218988617360571963085, −4.71434164784184326521767892969, −4.16354492780838585518091746458, −3.22572710599008572965081397253, −1.97721513367353748871179410879, −0.990964184631015222377669913319,
0.990964184631015222377669913319, 1.97721513367353748871179410879, 3.22572710599008572965081397253, 4.16354492780838585518091746458, 4.71434164784184326521767892969, 6.01234397218988617360571963085, 6.42497239306053748570404845338, 7.25266822370393144487599476460, 8.413818795997672639773834461783, 8.531058659433827241315008209209
