L(s) = 1 | + 3-s + 1.41·5-s − 1.41·7-s + 9-s + 1.41·15-s − 1.41·21-s + 1.00·25-s + 27-s − 1.41·29-s + 1.41·31-s − 2.00·35-s + 1.41·45-s + 1.00·49-s − 1.41·53-s − 2·59-s − 1.41·63-s + 1.00·75-s − 1.41·79-s + 81-s − 1.41·87-s + 1.41·93-s − 2·97-s + 1.41·101-s + 1.41·103-s − 2.00·105-s − 2·107-s + ⋯ |
L(s) = 1 | + 3-s + 1.41·5-s − 1.41·7-s + 9-s + 1.41·15-s − 1.41·21-s + 1.00·25-s + 27-s − 1.41·29-s + 1.41·31-s − 2.00·35-s + 1.41·45-s + 1.00·49-s − 1.41·53-s − 2·59-s − 1.41·63-s + 1.00·75-s − 1.41·79-s + 81-s − 1.41·87-s + 1.41·93-s − 2·97-s + 1.41·101-s + 1.41·103-s − 2.00·105-s − 2·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.712867688\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.712867688\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.550921394166160847716498469244, −9.176270549042840228222865053889, −8.176367708210517176732371886715, −7.16165400789348676851828029969, −6.39924665153750277548077557574, −5.77154670317581302555597347678, −4.53881449803914703632908118021, −3.36678728065054566479336907641, −2.67568195238321346961721808635, −1.63855927950080146808216236246,
1.63855927950080146808216236246, 2.67568195238321346961721808635, 3.36678728065054566479336907641, 4.53881449803914703632908118021, 5.77154670317581302555597347678, 6.39924665153750277548077557574, 7.16165400789348676851828029969, 8.176367708210517176732371886715, 9.176270549042840228222865053889, 9.550921394166160847716498469244