L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.00i·9-s + (1.41 + 1.41i)11-s − i·25-s + (0.707 + 0.707i)27-s − 2.00·33-s + 49-s + (1.41 + 1.41i)59-s + 2i·73-s + (0.707 + 0.707i)75-s − 1.00·81-s + (−1.41 + 1.41i)83-s + 2·97-s + (1.41 − 1.41i)99-s + (−1.41 − 1.41i)107-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.00i·9-s + (1.41 + 1.41i)11-s − i·25-s + (0.707 + 0.707i)27-s − 2.00·33-s + 49-s + (1.41 + 1.41i)59-s + 2i·73-s + (0.707 + 0.707i)75-s − 1.00·81-s + (−1.41 + 1.41i)83-s + 2·97-s + (1.41 − 1.41i)99-s + (−1.41 − 1.41i)107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.038388125\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038388125\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 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
−9.143587767892117057467546409334, −8.501876797379283641808797221342, −7.26193297399623830351969051673, −6.75695514858463243316279194096, −5.99574931940502672455653576533, −5.12663337476553957180311472218, −4.24438376670352310000488130924, −3.90294273121176848437951031786, −2.50229553473516403073351853980, −1.21653681652877569593334382239,
0.848209316173399195777955746979, 1.83951326644066734521503688133, 3.15747861379236883038058783961, 4.00689566264957394810232429056, 5.10259930224557558556830228620, 5.86135806230783521844654220983, 6.43158330850156362504776652469, 7.11794341635796026076304256279, 7.943758493597920639692317887076, 8.732914256296874498474520156530