| L(s) = 1 | + 1.36·2-s − 9.26·3-s − 6.14·4-s − 1.21·5-s − 12.6·6-s − 10.5·7-s − 19.2·8-s + 58.8·9-s − 1.65·10-s + 39.9·11-s + 56.8·12-s − 52.4·13-s − 14.3·14-s + 11.2·15-s + 22.8·16-s + 1.95·17-s + 80.1·18-s + 86.6·19-s + 7.46·20-s + 97.5·21-s + 54.4·22-s − 205.·23-s + 178.·24-s − 123.·25-s − 71.5·26-s − 294.·27-s + 64.6·28-s + ⋯ |
| L(s) = 1 | + 0.482·2-s − 1.78·3-s − 0.767·4-s − 0.108·5-s − 0.859·6-s − 0.568·7-s − 0.852·8-s + 2.17·9-s − 0.0524·10-s + 1.09·11-s + 1.36·12-s − 1.11·13-s − 0.274·14-s + 0.193·15-s + 0.356·16-s + 0.0279·17-s + 1.05·18-s + 1.04·19-s + 0.0834·20-s + 1.01·21-s + 0.527·22-s − 1.86·23-s + 1.51·24-s − 0.988·25-s − 0.539·26-s − 2.10·27-s + 0.436·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2784754333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2784754333\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.36T + 8T^{2} \) |
| 3 | \( 1 + 9.26T + 27T^{2} \) |
| 5 | \( 1 + 1.21T + 125T^{2} \) |
| 7 | \( 1 + 10.5T + 343T^{2} \) |
| 11 | \( 1 - 39.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.95T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 30.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 38.7T + 6.89e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 71.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 1.90T + 2.26e5T^{2} \) |
| 67 | \( 1 + 925.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 877.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 661.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 103.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3T + 9.12e5T^{2} \) |
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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
−9.189158223982416627444993259989, −7.908516258838667462447317223973, −7.01382433438892213116963833610, −6.14830166551873870760999006631, −5.75806027765207787147688847584, −4.78875582530884420131377221425, −4.26938334978359979564652294885, −3.30133490632772537936586107027, −1.51880622031549640112221730357, −0.26304167224510463479003525241,
0.26304167224510463479003525241, 1.51880622031549640112221730357, 3.30133490632772537936586107027, 4.26938334978359979564652294885, 4.78875582530884420131377221425, 5.75806027765207787147688847584, 6.14830166551873870760999006631, 7.01382433438892213116963833610, 7.908516258838667462447317223973, 9.189158223982416627444993259989
