| L(s) = 1 | + 0.911·3-s + 2.21·5-s + 0.592·7-s − 26.1·9-s − 44.0·11-s − 39.4·13-s + 2.01·15-s − 89.6·17-s + 15.7·19-s + 0.539·21-s + 23·23-s − 120.·25-s − 48.4·27-s + 227.·29-s − 83.0·31-s − 40.1·33-s + 1.31·35-s + 201.·37-s − 35.9·39-s + 364.·41-s + 79.8·43-s − 57.9·45-s + 162.·47-s − 342.·49-s − 81.7·51-s + 637.·53-s − 97.5·55-s + ⋯ |
| L(s) = 1 | + 0.175·3-s + 0.198·5-s + 0.0319·7-s − 0.969·9-s − 1.20·11-s − 0.842·13-s + 0.0347·15-s − 1.27·17-s + 0.190·19-s + 0.00560·21-s + 0.208·23-s − 0.960·25-s − 0.345·27-s + 1.45·29-s − 0.481·31-s − 0.211·33-s + 0.00633·35-s + 0.894·37-s − 0.147·39-s + 1.38·41-s + 0.283·43-s − 0.191·45-s + 0.503·47-s − 0.998·49-s − 0.224·51-s + 1.65·53-s − 0.239·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.315032864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315032864\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 0.911T + 27T^{2} \) |
| 5 | \( 1 - 2.21T + 125T^{2} \) |
| 7 | \( 1 - 0.592T + 343T^{2} \) |
| 11 | \( 1 + 44.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 89.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 227.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 83.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 79.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 162.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 637.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 830.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 23.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 689.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 701.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 332.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 208.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 93.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 290.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 721.T + 9.12e5T^{2} \) |
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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
−9.115803301402214630317488840351, −8.307190455029228026862074213913, −7.66023164364014308860562623878, −6.71605605348518623918859785397, −5.74530731805368169213196940810, −5.06749204524607896649303798313, −4.07948728110511268267490364783, −2.69459725071902991609383646993, −2.32507268742733253954946052997, −0.52320242245184258425608438634,
0.52320242245184258425608438634, 2.32507268742733253954946052997, 2.69459725071902991609383646993, 4.07948728110511268267490364783, 5.06749204524607896649303798313, 5.74530731805368169213196940810, 6.71605605348518623918859785397, 7.66023164364014308860562623878, 8.307190455029228026862074213913, 9.115803301402214630317488840351
