| L(s) = 1 | − 2.47·2-s + 4.11·4-s + 4.22·5-s + 1.64·7-s − 5.22·8-s − 10.4·10-s + 2.35·11-s − 2.58·13-s − 4.06·14-s + 4.70·16-s − 5.87·17-s + 2.94·19-s + 17.4·20-s − 5.83·22-s + 2.22·23-s + 12.8·25-s + 6.39·26-s + 6.75·28-s − 29-s − 2.22·31-s − 1.16·32-s + 14.5·34-s + 6.94·35-s − 0.945·37-s − 7.28·38-s − 22.1·40-s − 0.568·41-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.05·4-s + 1.89·5-s + 0.620·7-s − 1.84·8-s − 3.30·10-s + 0.710·11-s − 0.717·13-s − 1.08·14-s + 1.17·16-s − 1.42·17-s + 0.675·19-s + 3.89·20-s − 1.24·22-s + 0.464·23-s + 2.57·25-s + 1.25·26-s + 1.27·28-s − 0.185·29-s − 0.400·31-s − 0.206·32-s + 2.49·34-s + 1.17·35-s − 0.155·37-s − 1.18·38-s − 3.49·40-s − 0.0887·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8276697708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8276697708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.945T + 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 7.28T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 - 0.926T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 0.945T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−11.46124877115121423737375890435, −10.74080013280750863418459716338, −9.735640367801854909548350458050, −9.309956059886718691277450343830, −8.464601111909517501056914192685, −7.11114218892230118223003304842, −6.37470607736323218614690014798, −5.07477201768043971528596570792, −2.43102823933071948428892909782, −1.49442799016086090491750613038,
1.49442799016086090491750613038, 2.43102823933071948428892909782, 5.07477201768043971528596570792, 6.37470607736323218614690014798, 7.11114218892230118223003304842, 8.464601111909517501056914192685, 9.309956059886718691277450343830, 9.735640367801854909548350458050, 10.74080013280750863418459716338, 11.46124877115121423737375890435
