| L(s) = 1 | + 2·3-s − 3.46·5-s − 4.73·7-s + 9-s + 1.26·11-s − 13-s − 6.92·15-s − 1.46·17-s − 2.73·19-s − 9.46·21-s + 4·23-s + 6.99·25-s − 4·27-s − 2·29-s + 3.26·31-s + 2.53·33-s + 16.3·35-s + 4.92·37-s − 2·39-s + 4.92·41-s + 7.46·43-s − 3.46·45-s − 3.26·47-s + 15.3·49-s − 2.92·51-s + 10.9·53-s − 4.39·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.54·5-s − 1.78·7-s + 0.333·9-s + 0.382·11-s − 0.277·13-s − 1.78·15-s − 0.355·17-s − 0.626·19-s − 2.06·21-s + 0.834·23-s + 1.39·25-s − 0.769·27-s − 0.371·29-s + 0.586·31-s + 0.441·33-s + 2.77·35-s + 0.810·37-s − 0.320·39-s + 0.769·41-s + 1.13·43-s − 0.516·45-s − 0.476·47-s + 2.19·49-s − 0.410·51-s + 1.50·53-s − 0.592·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187845093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187845093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 6.73T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.622076288579392155964461226193, −7.975975262870032889106454267237, −7.17236414352687234796768068874, −6.71240091680589272447303787065, −5.70495272670373130184862222283, −4.26181703856032238634869666773, −3.87633909156815732243980757314, −3.06324705379444144622456790442, −2.50955288530124832694397755115, −0.58137463746360388236987393310,
0.58137463746360388236987393310, 2.50955288530124832694397755115, 3.06324705379444144622456790442, 3.87633909156815732243980757314, 4.26181703856032238634869666773, 5.70495272670373130184862222283, 6.71240091680589272447303787065, 7.17236414352687234796768068874, 7.975975262870032889106454267237, 8.622076288579392155964461226193
