| L(s) = 1 | − 4.55·2-s + 3·3-s + 12.7·4-s − 17.8·5-s − 13.6·6-s − 21.6·8-s + 9·9-s + 81.4·10-s − 11.3·11-s + 38.2·12-s + 13.0·13-s − 53.6·15-s − 3.25·16-s − 53.2·17-s − 41.0·18-s + 42.4·19-s − 228.·20-s + 51.9·22-s + 152.·23-s − 65.0·24-s + 194.·25-s − 59.6·26-s + 27·27-s + 186.·29-s + 244.·30-s + 157.·31-s + 188.·32-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.59·4-s − 1.59·5-s − 0.930·6-s − 0.958·8-s + 0.333·9-s + 2.57·10-s − 0.312·11-s + 0.920·12-s + 0.279·13-s − 0.922·15-s − 0.0508·16-s − 0.759·17-s − 0.536·18-s + 0.512·19-s − 2.54·20-s + 0.503·22-s + 1.37·23-s − 0.553·24-s + 1.55·25-s − 0.450·26-s + 0.192·27-s + 1.19·29-s + 1.48·30-s + 0.914·31-s + 1.04·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6305026309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6305026309\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.55T + 8T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 11 | \( 1 + 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 42.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 186.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 3.74T + 5.06e4T^{2} \) |
| 41 | \( 1 - 39.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 21.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 365.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 651.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 368.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 608.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 910.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 327.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 37.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 722.T + 9.12e5T^{2} \) |
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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
−12.21261240862741050813824339180, −11.25103727768136069890863980838, −10.49264160024120041925105176204, −9.178044932487093251917151477814, −8.436575154599600802760781964993, −7.66946559103578017723647040468, −6.84605740364335760849466980946, −4.43942165374457346908375915859, −2.86109767915689843554766852506, −0.805453879767246941485965050539,
0.805453879767246941485965050539, 2.86109767915689843554766852506, 4.43942165374457346908375915859, 6.84605740364335760849466980946, 7.66946559103578017723647040468, 8.436575154599600802760781964993, 9.178044932487093251917151477814, 10.49264160024120041925105176204, 11.25103727768136069890863980838, 12.21261240862741050813824339180
