| L(s) = 1 | − 13.2·5-s − 5.23·7-s + 11-s − 84.9·13-s − 40.9·17-s − 57.5·19-s − 114.·23-s + 50.1·25-s − 202.·29-s + 274.·31-s + 69.2·35-s − 242.·37-s + 328.·41-s − 281.·43-s − 23.8·47-s − 315.·49-s + 300.·53-s − 13.2·55-s + 753.·59-s − 495.·61-s + 1.12e3·65-s + 409.·67-s − 1.11e3·71-s − 287·73-s − 5.23·77-s − 1.23e3·79-s − 942.·83-s + ⋯ |
| L(s) = 1 | − 1.18·5-s − 0.282·7-s + 0.0274·11-s − 1.81·13-s − 0.584·17-s − 0.694·19-s − 1.04·23-s + 0.401·25-s − 1.29·29-s + 1.58·31-s + 0.334·35-s − 1.07·37-s + 1.25·41-s − 0.997·43-s − 0.0740·47-s − 0.920·49-s + 0.777·53-s − 0.0324·55-s + 1.66·59-s − 1.03·61-s + 2.14·65-s + 0.747·67-s − 1.86·71-s − 0.460·73-s − 0.00774·77-s − 1.75·79-s − 1.24·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3488473570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3488473570\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 13.2T + 125T^{2} \) |
| 7 | \( 1 + 5.23T + 343T^{2} \) |
| 11 | \( 1 - T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 409.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 942.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 190.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 306.T + 9.12e5T^{2} \) |
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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
−8.852598961083197197528789846046, −8.093538476940265831923523915758, −7.39875492841859669811307863933, −6.77603529730494763106951721783, −5.70481139395128206556376314217, −4.60850188533501466780050058982, −4.08959380001062514371271192783, −2.98438216033985609552491641886, −2.00385119534232641666693405488, −0.26302693884667498888440145457,
0.26302693884667498888440145457, 2.00385119534232641666693405488, 2.98438216033985609552491641886, 4.08959380001062514371271192783, 4.60850188533501466780050058982, 5.70481139395128206556376314217, 6.77603529730494763106951721783, 7.39875492841859669811307863933, 8.093538476940265831923523915758, 8.852598961083197197528789846046
