| L(s) = 1 | + 6·5-s − 21.1·7-s + 42.3·11-s − 20·13-s + 8·17-s − 84.6·19-s + 169.·23-s − 89·25-s − 46·29-s − 21.1·31-s − 126.·35-s + 164·37-s + 312·41-s − 423.·43-s + 169.·47-s + 105.·49-s + 266·53-s + 253.·55-s − 253.·59-s + 132·61-s − 120·65-s − 507.·67-s − 677.·71-s + 246·73-s − 896.·77-s + 232.·79-s + 973.·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 1.14·7-s + 1.16·11-s − 0.426·13-s + 0.114·17-s − 1.02·19-s + 1.53·23-s − 0.711·25-s − 0.294·29-s − 0.122·31-s − 0.613·35-s + 0.728·37-s + 1.18·41-s − 1.50·43-s + 0.525·47-s + 0.306·49-s + 0.689·53-s + 0.622·55-s − 0.560·59-s + 0.277·61-s − 0.228·65-s − 0.926·67-s − 1.13·71-s + 0.394·73-s − 1.32·77-s + 0.331·79-s + 1.28·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.974917987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.974917987\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6T + 125T^{2} \) |
| 7 | \( 1 + 21.1T + 343T^{2} \) |
| 11 | \( 1 - 42.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 46T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 164T + 5.06e4T^{2} \) |
| 41 | \( 1 - 312T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 266T + 1.48e5T^{2} \) |
| 59 | \( 1 + 253.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 132T + 2.26e5T^{2} \) |
| 67 | \( 1 + 507.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 677.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 246T + 3.89e5T^{2} \) |
| 79 | \( 1 - 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 973.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 302T + 9.12e5T^{2} \) |
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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
−9.068320718430586707576346592331, −7.82914768500903149922454862135, −6.89088889203380375911102492517, −6.39493412954873162441148636136, −5.69092748653825105772375589094, −4.60339266337629697763503323342, −3.72186251088187267584349983865, −2.84899815356566449971456843329, −1.82057798586685079973966681706, −0.62323311141488764755729211682,
0.62323311141488764755729211682, 1.82057798586685079973966681706, 2.84899815356566449971456843329, 3.72186251088187267584349983865, 4.60339266337629697763503323342, 5.69092748653825105772375589094, 6.39493412954873162441148636136, 6.89088889203380375911102492517, 7.82914768500903149922454862135, 9.068320718430586707576346592331
