| L(s) = 1 | + 2.06·2-s + 2.27·4-s + 0.238·5-s + 7-s + 0.558·8-s + 0.491·10-s + 5.40·11-s + 0.238·13-s + 2.06·14-s − 3.38·16-s + 17-s + 7.89·19-s + 0.540·20-s + 11.1·22-s − 6.38·23-s − 4.94·25-s + 0.491·26-s + 2.27·28-s − 4.13·29-s + 6.18·31-s − 8.11·32-s + 2.06·34-s + 0.238·35-s + 5.64·37-s + 16.3·38-s + 0.132·40-s + 6.30·41-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 1.13·4-s + 0.106·5-s + 0.377·7-s + 0.197·8-s + 0.155·10-s + 1.62·11-s + 0.0660·13-s + 0.552·14-s − 0.846·16-s + 0.242·17-s + 1.81·19-s + 0.120·20-s + 2.38·22-s − 1.33·23-s − 0.988·25-s + 0.0964·26-s + 0.429·28-s − 0.767·29-s + 1.11·31-s − 1.43·32-s + 0.354·34-s + 0.0402·35-s + 0.927·37-s + 2.64·38-s + 0.0210·40-s + 0.984·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.895887888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.895887888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.06T + 2T^{2} \) |
| 5 | \( 1 - 0.238T + 5T^{2} \) |
| 11 | \( 1 - 5.40T + 11T^{2} \) |
| 13 | \( 1 - 0.238T + 13T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 - 6.18T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 6.30T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 6.18T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−9.739577470556639729471412910055, −9.299429622962074610599884804409, −8.028469031447071920982617747618, −7.16072200889284526471138682033, −6.07925529952524005612740811646, −5.69809697209128229580741345844, −4.45274646811917775028645126182, −3.91856146993491953860915128143, −2.87391141196568676495496466894, −1.47870298937905441913389595495,
1.47870298937905441913389595495, 2.87391141196568676495496466894, 3.91856146993491953860915128143, 4.45274646811917775028645126182, 5.69809697209128229580741345844, 6.07925529952524005612740811646, 7.16072200889284526471138682033, 8.028469031447071920982617747618, 9.299429622962074610599884804409, 9.739577470556639729471412910055
