L(s) = 1 | + 1.35i·2-s + 2.17·4-s + 5.10i·5-s + 13.1·7-s + 8.34i·8-s − 6.89·10-s + 2.67·13-s + 17.7i·14-s − 2.58·16-s + 17.2i·17-s − 30.6·19-s + 11.0i·20-s + 15.5i·23-s − 1.05·25-s + 3.60i·26-s + ⋯ |
L(s) = 1 | + 0.675i·2-s + 0.543·4-s + 1.02i·5-s + 1.88·7-s + 1.04i·8-s − 0.689·10-s + 0.205·13-s + 1.27i·14-s − 0.161·16-s + 1.01i·17-s − 1.61·19-s + 0.554i·20-s + 0.675i·23-s − 0.0423·25-s + 0.138i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.871309201\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.871309201\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.35iT - 4T^{2} \) |
| 5 | \( 1 - 5.10iT - 25T^{2} \) |
| 7 | \( 1 - 13.1T + 49T^{2} \) |
| 13 | \( 1 - 2.67T + 169T^{2} \) |
| 17 | \( 1 - 17.2iT - 289T^{2} \) |
| 19 | \( 1 + 30.6T + 361T^{2} \) |
| 23 | \( 1 - 15.5iT - 529T^{2} \) |
| 29 | \( 1 + 10.3iT - 841T^{2} \) |
| 31 | \( 1 - 16.8T + 961T^{2} \) |
| 37 | \( 1 - 18.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 82.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 60.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 49.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 6.37T + 5.32e3T^{2} \) |
| 79 | \( 1 + 115.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 40.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 71.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 18.6T + 9.40e3T^{2} \) |
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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
−10.34696238326814927221200484871, −8.658770998310431243102709706144, −8.202095131185221681245305978162, −7.45665275332242933522238670575, −6.64987636092397530060790298708, −5.90170473288643114220456330628, −4.95268018668401692446157194941, −3.88776444593811265976034702938, −2.42357294598628262600914284141, −1.68397407304568711660063459454,
0.884757922641631687962485860169, 1.73019590720618898092452401230, 2.71050383928338132654529605895, 4.35534492494196345319337716468, 4.69744171377640338283760951123, 5.86322704122477556047917051632, 6.96060610422878705825377242145, 7.962344558527300778847270878045, 8.525538968752196382092538368966, 9.374540801969055235748606081904