| L(s) = 1 | + 0.688·3-s − 5-s + 7-s − 2.52·9-s + 11-s − 3.73·13-s − 0.688·15-s + 0.0666·17-s − 6.42·19-s + 0.688·21-s + 1.09·23-s + 25-s − 3.80·27-s − 7.80·29-s + 5.59·31-s + 0.688·33-s − 35-s + 1.33·37-s − 2.57·39-s + 6.64·41-s + 11.7·43-s + 2.52·45-s + 2.26·47-s + 49-s + 0.0459·51-s − 1.71·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.397·3-s − 0.447·5-s + 0.377·7-s − 0.841·9-s + 0.301·11-s − 1.03·13-s − 0.177·15-s + 0.0161·17-s − 1.47·19-s + 0.150·21-s + 0.228·23-s + 0.200·25-s − 0.732·27-s − 1.44·29-s + 1.00·31-s + 0.119·33-s − 0.169·35-s + 0.218·37-s − 0.412·39-s + 1.03·41-s + 1.79·43-s + 0.376·45-s + 0.329·47-s + 0.142·49-s + 0.00643·51-s − 0.236·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545333477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545333477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 0.688T + 3T^{2} \) |
| 13 | \( 1 + 3.73T + 13T^{2} \) |
| 17 | \( 1 - 0.0666T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 - 1.09T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.915117334290678257209378764128, −7.62349634109667120410982152085, −6.67442874345731022763333283237, −5.94719222680631447334436350140, −5.14214221989441657571192997040, −4.33404123929775020782447085075, −3.70599861949389155592111944829, −2.62355682720269459118567804941, −2.11627276282560302254353628990, −0.60980537183279820986034420748,
0.60980537183279820986034420748, 2.11627276282560302254353628990, 2.62355682720269459118567804941, 3.70599861949389155592111944829, 4.33404123929775020782447085075, 5.14214221989441657571192997040, 5.94719222680631447334436350140, 6.67442874345731022763333283237, 7.62349634109667120410982152085, 7.915117334290678257209378764128
