L(s) = 1 | − 7.73·3-s − 5·5-s − 23.5·7-s + 32.7·9-s + 32.1·11-s + 40.0·13-s + 38.6·15-s + 126.·17-s − 0.232·19-s + 182.·21-s + 23·23-s + 25·25-s − 44.5·27-s − 137.·29-s − 112.·31-s − 248.·33-s + 117.·35-s + 45.7·37-s − 309.·39-s − 135.·41-s − 543.·43-s − 163.·45-s − 26.4·47-s + 212.·49-s − 975.·51-s + 43.6·53-s − 160.·55-s + ⋯ |
L(s) = 1 | − 1.48·3-s − 0.447·5-s − 1.27·7-s + 1.21·9-s + 0.880·11-s + 0.855·13-s + 0.665·15-s + 1.79·17-s − 0.00281·19-s + 1.89·21-s + 0.208·23-s + 0.200·25-s − 0.317·27-s − 0.878·29-s − 0.653·31-s − 1.31·33-s + 0.568·35-s + 0.203·37-s − 1.27·39-s − 0.515·41-s − 1.92·43-s − 0.542·45-s − 0.0820·47-s + 0.618·49-s − 2.67·51-s + 0.113·53-s − 0.393·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8045128434\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8045128434\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
good | 3 | \( 1 + 7.73T + 27T^{2} \) |
| 7 | \( 1 + 23.5T + 343T^{2} \) |
| 11 | \( 1 - 32.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.232T + 6.85e3T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 45.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 543.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 26.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 43.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 202.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 150.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 667.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 602.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 485.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.060028897535342528541398842910, −7.986083057677233223276951786722, −6.99174934272841970700791005633, −6.43281136248278194634687419635, −5.77886825034172779044131300122, −5.04413336384957841475612249174, −3.77826805176956558262818533438, −3.32705388150961290378074526129, −1.41878223218199821610406992081, −0.49462493273344896557532024330,
0.49462493273344896557532024330, 1.41878223218199821610406992081, 3.32705388150961290378074526129, 3.77826805176956558262818533438, 5.04413336384957841475612249174, 5.77886825034172779044131300122, 6.43281136248278194634687419635, 6.99174934272841970700791005633, 7.986083057677233223276951786722, 9.060028897535342528541398842910