| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s − 6·17-s + 23-s + 25-s − 27-s + 6·29-s − 6·37-s + 2·39-s − 6·41-s − 45-s + 8·47-s − 7·49-s + 6·51-s − 6·53-s + 12·59-s + 10·61-s + 2·65-s − 69-s − 6·73-s − 75-s − 8·79-s + 81-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.149·45-s + 1.16·47-s − 49-s + 0.840·51-s − 0.824·53-s + 1.56·59-s + 1.28·61-s + 0.248·65-s − 0.120·69-s − 0.702·73-s − 0.115·75-s − 0.900·79-s + 1/9·81-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9660884825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9660884825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.261857922282941557718363886437, −7.17956093738550277882374666377, −6.86350897614877372587242381986, −6.07005303301742810983912044735, −5.11865575551650886603344632437, −4.61468691486900089813200151485, −3.81642148867192917219560574909, −2.80414961451346977836271204726, −1.84200022138086438290643263532, −0.52992735372054663786949256501,
0.52992735372054663786949256501, 1.84200022138086438290643263532, 2.80414961451346977836271204726, 3.81642148867192917219560574909, 4.61468691486900089813200151485, 5.11865575551650886603344632437, 6.07005303301742810983912044735, 6.86350897614877372587242381986, 7.17956093738550277882374666377, 8.261857922282941557718363886437
