L(s) = 1 | + (−1.41 − 1.41i)3-s + 3.00i·9-s + (2.82 − 2.82i)27-s + 2·41-s + (1.41 + 1.41i)43-s − i·49-s + (1.41 − 1.41i)67-s − 5.00·81-s + (−1.41 − 1.41i)83-s − 2i·89-s + (−1.41 + 1.41i)107-s + ⋯ |
L(s) = 1 | + (−1.41 − 1.41i)3-s + 3.00i·9-s + (2.82 − 2.82i)27-s + 2·41-s + (1.41 + 1.41i)43-s − i·49-s + (1.41 − 1.41i)67-s − 5.00·81-s + (−1.41 − 1.41i)83-s − 2i·89-s + (−1.41 + 1.41i)107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7171609862\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7171609862\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 2T + T^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.394980667452418383204253627839, −7.67227275574834980072981229281, −7.19149568411239658681494494036, −6.32007069002753932815979250103, −5.89256752230771352010204573588, −5.08099743099059391598747660270, −4.28700261138015981597641656976, −2.73645256062189889466431933232, −1.80342857573806925923733378552, −0.74543994822053037423724224805,
0.902737017270382947585579771008, 2.74304594496652906932792203210, 3.94058827844325653207229184536, 4.27839128249732722841584331777, 5.32560041475320584520837822407, 5.72297536587405957385307798674, 6.53136495634829066566097288440, 7.30708584688644988817627490885, 8.494015596122020253857576034630, 9.345792579074569173699567312199