L(s) = 1 | + (−0.5 − 0.866i)2-s + (3.5 − 6.06i)4-s + (−2.5 + 4.33i)5-s + (3 + 5.19i)7-s − 15·8-s + 5·10-s + (23.5 + 40.7i)11-s + (2.5 − 4.33i)13-s + (3 − 5.19i)14-s + (−20.5 − 35.5i)16-s − 131·17-s − 56·19-s + (17.5 + 30.3i)20-s + (23.5 − 40.7i)22-s + (−1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.176 − 0.306i)2-s + (0.437 − 0.757i)4-s + (−0.223 + 0.387i)5-s + (0.161 + 0.280i)7-s − 0.662·8-s + 0.158·10-s + (0.644 + 1.11i)11-s + (0.0533 − 0.0923i)13-s + (0.0572 − 0.0991i)14-s + (−0.320 − 0.554i)16-s − 1.86·17-s − 0.676·19-s + (0.195 + 0.338i)20-s + (0.227 − 0.394i)22-s + (−0.0135 + 0.0235i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7718896226\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7718896226\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-3 - 5.19i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-23.5 - 40.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 131T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56T + 6.85e3T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-78.5 - 135. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (112.5 - 194. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 70T + 5.06e4T^{2} \) |
| 41 | \( 1 + (70 - 121. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (198.5 + 343. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-173.5 - 300. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 4T + 1.48e5T^{2} \) |
| 59 | \( 1 + (374 - 647. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-169 - 292. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (246 - 426. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 32T + 3.57e5T^{2} \) |
| 73 | \( 1 - 970T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-628.5 - 1.08e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-51 - 88.3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (487 + 843. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.92244342972940366868759375716, −10.40352977389471248843630096273, −9.299338505080798643872617590484, −8.604770110151352397062809517034, −6.97229033533733779407469331485, −6.65491115174719910200172102357, −5.28601063054817385014137866245, −4.17242442174866633496967974189, −2.57493960609272390675170464096, −1.59904946002447965331617038683,
0.25420302159664218699094324509, 2.15036841508936084463792814075, 3.58997144572511876176638193493, 4.48945727991561169321278616129, 6.11274720338049867790055534378, 6.76748556731775105490301061839, 7.941206292556602256481770257707, 8.606373771847465909485087995233, 9.340219338216906302378144865770, 10.94308716979438142381523081130