| L(s) = 1 | + 3·3-s − 20.1·5-s − 19.6·7-s + 9·9-s + 13.5·11-s + 13·13-s − 60.4·15-s + 116.·17-s + 68.1·19-s − 59.0·21-s + 122.·23-s + 280.·25-s + 27·27-s + 204.·29-s − 194.·31-s + 40.6·33-s + 396.·35-s − 142.·37-s + 39·39-s − 175.·41-s − 219.·43-s − 181.·45-s + 236.·47-s + 45.0·49-s + 349.·51-s − 628.·53-s − 273.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.80·5-s − 1.06·7-s + 0.333·9-s + 0.371·11-s + 0.277·13-s − 1.04·15-s + 1.66·17-s + 0.822·19-s − 0.614·21-s + 1.10·23-s + 2.24·25-s + 0.192·27-s + 1.30·29-s − 1.12·31-s + 0.214·33-s + 1.91·35-s − 0.634·37-s + 0.160·39-s − 0.666·41-s − 0.778·43-s − 0.600·45-s + 0.733·47-s + 0.131·49-s + 0.958·51-s − 1.62·53-s − 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.455150405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455150405\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 20.1T + 125T^{2} \) |
| 7 | \( 1 + 19.6T + 343T^{2} \) |
| 11 | \( 1 - 13.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 68.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 219.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 236.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 224.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 902.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 15.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 670.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 562.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−11.39429318280176290491104612639, −10.26319618218411838502990788453, −9.261487560716465069374359732782, −8.305392067869199820829314185354, −7.48283110829867664917863584212, −6.67751471889653331595213889986, −5.02071010652252472414498437730, −3.52634100507530736562074259558, −3.30149869625459484545763770135, −0.821943270193890418587602049894,
0.821943270193890418587602049894, 3.30149869625459484545763770135, 3.52634100507530736562074259558, 5.02071010652252472414498437730, 6.67751471889653331595213889986, 7.48283110829867664917863584212, 8.305392067869199820829314185354, 9.261487560716465069374359732782, 10.26319618218411838502990788453, 11.39429318280176290491104612639
