L(s) = 1 | + 3.22·5-s + 6.57i·7-s − 13.8i·11-s + 19.0·13-s + 23.0·17-s + 18.8i·19-s − 13.5i·23-s − 14.5·25-s − 0.752·29-s − 46.4i·31-s + 21.1i·35-s + 2.68·37-s − 34.1·41-s + 20.9i·43-s + 15.4i·47-s + ⋯ |
L(s) = 1 | + 0.645·5-s + 0.938i·7-s − 1.26i·11-s + 1.46·13-s + 1.35·17-s + 0.991i·19-s − 0.591i·23-s − 0.583·25-s − 0.0259·29-s − 1.49i·31-s + 0.605i·35-s + 0.0724·37-s − 0.832·41-s + 0.486i·43-s + 0.327i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.622884156\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.622884156\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.22T + 25T^{2} \) |
| 7 | \( 1 - 6.57iT - 49T^{2} \) |
| 11 | \( 1 + 13.8iT - 121T^{2} \) |
| 13 | \( 1 - 19.0T + 169T^{2} \) |
| 17 | \( 1 - 23.0T + 289T^{2} \) |
| 19 | \( 1 - 18.8iT - 361T^{2} \) |
| 23 | \( 1 + 13.5iT - 529T^{2} \) |
| 29 | \( 1 + 0.752T + 841T^{2} \) |
| 31 | \( 1 + 46.4iT - 961T^{2} \) |
| 37 | \( 1 - 2.68T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 15.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 40.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 169.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 93.1T + 9.40e3T^{2} \) |
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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
−9.050708969033842079936862523576, −8.385696222598398024817914297515, −7.82407718805822463333476029701, −6.36754620065328886246644342415, −5.85383860535186867872473527473, −5.46259332509171071946703896832, −3.92756588715366879696389115088, −3.18003661423528055504375555462, −2.03671156391229720273808163512, −0.908135406409234137233289337103,
0.988435413613614603708136115045, 1.86830006350095114456484087911, 3.26473832329013798222392096743, 4.06926109197254529311945892143, 5.09424491280942415794320141542, 5.87178827999754965947634884051, 6.91152285145126010103021990712, 7.35376643737461280990565793686, 8.397551191950960898691847069455, 9.184690591718623962730765471229