| L(s) = 1 | + 3-s − 5-s − 0.681·7-s + 9-s − 3.18·11-s + 13-s − 15-s + 3.69·17-s − 3.87·19-s − 0.681·21-s − 6.04·23-s + 25-s + 27-s + 4.85·29-s + 4·31-s − 3.18·33-s + 0.681·35-s + 5.18·37-s + 39-s + 1.18·41-s − 6.37·43-s − 45-s + 10.7·47-s − 6.53·49-s + 3.69·51-s − 7.56·53-s + 3.18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.257·7-s + 0.333·9-s − 0.961·11-s + 0.277·13-s − 0.258·15-s + 0.896·17-s − 0.888·19-s − 0.148·21-s − 1.26·23-s + 0.200·25-s + 0.192·27-s + 0.901·29-s + 0.718·31-s − 0.555·33-s + 0.115·35-s + 0.853·37-s + 0.160·39-s + 0.185·41-s − 0.972·43-s − 0.149·45-s + 1.56·47-s − 0.933·49-s + 0.517·51-s − 1.03·53-s + 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877445954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877445954\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 0.681T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 1.82T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−8.083853776912826161464647686644, −7.59551242553866384787397713547, −6.62036679241614847665031389416, −6.03640580424431193897516602648, −5.08634210236837009596409191762, −4.32859579416541790719222521348, −3.56902569332840156590365931314, −2.81429257764261978914754210973, −2.00662359035943474790942900837, −0.67878950524728812232264579612,
0.67878950524728812232264579612, 2.00662359035943474790942900837, 2.81429257764261978914754210973, 3.56902569332840156590365931314, 4.32859579416541790719222521348, 5.08634210236837009596409191762, 6.03640580424431193897516602648, 6.62036679241614847665031389416, 7.59551242553866384787397713547, 8.083853776912826161464647686644
