| L(s) = 1 | − 2.77·3-s − 14.3·5-s + 16.7·7-s − 19.3·9-s + 10.9·11-s − 13·13-s + 39.6·15-s + 74.6·17-s + 27.8·19-s − 46.4·21-s + 137.·23-s + 79.9·25-s + 128.·27-s + 166.·29-s + 184.·31-s − 30.2·33-s − 240.·35-s − 188.·37-s + 36.0·39-s + 450.·41-s − 456.·43-s + 276.·45-s + 139.·47-s − 61.6·49-s − 206.·51-s + 290.·53-s − 156.·55-s + ⋯ |
| L(s) = 1 | − 0.533·3-s − 1.28·5-s + 0.905·7-s − 0.715·9-s + 0.299·11-s − 0.277·13-s + 0.683·15-s + 1.06·17-s + 0.335·19-s − 0.483·21-s + 1.24·23-s + 0.639·25-s + 0.915·27-s + 1.06·29-s + 1.06·31-s − 0.159·33-s − 1.15·35-s − 0.838·37-s + 0.147·39-s + 1.71·41-s − 1.61·43-s + 0.916·45-s + 0.434·47-s − 0.179·49-s − 0.568·51-s + 0.752·53-s − 0.382·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.157428264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157428264\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 + 2.77T + 27T^{2} \) |
| 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 - 16.7T + 343T^{2} \) |
| 11 | \( 1 - 10.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 74.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 450.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 30.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 275.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 191.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 859.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 345.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 925.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 276.T + 9.12e5T^{2} \) |
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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
−11.69934353588118925315806308986, −11.34448533203316343858440427760, −10.20653756175255871479292642485, −8.682651380211539059415561024589, −7.960760640795225794427584165473, −6.92932776990300621166014870186, −5.46466021012124351503259497773, −4.50380591524760242079717165695, −3.09587395373269998577770677773, −0.855680583671635048168810857721,
0.855680583671635048168810857721, 3.09587395373269998577770677773, 4.50380591524760242079717165695, 5.46466021012124351503259497773, 6.92932776990300621166014870186, 7.960760640795225794427584165473, 8.682651380211539059415561024589, 10.20653756175255871479292642485, 11.34448533203316343858440427760, 11.69934353588118925315806308986
