L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.965 − 0.258i)3-s + (−0.499 + 0.866i)4-s + (0.707 + 0.707i)5-s + (0.258 + 0.965i)6-s + (−0.866 − 0.5i)7-s + 0.999·8-s + (0.258 − 0.965i)10-s + (0.707 − 0.707i)12-s + (0.258 + 0.965i)13-s + 0.999i·14-s + (−0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)20-s + (0.707 + 0.707i)21-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.965 − 0.258i)3-s + (−0.499 + 0.866i)4-s + (0.707 + 0.707i)5-s + (0.258 + 0.965i)6-s + (−0.866 − 0.5i)7-s + 0.999·8-s + (0.258 − 0.965i)10-s + (0.707 − 0.707i)12-s + (0.258 + 0.965i)13-s + 0.999i·14-s + (−0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)20-s + (0.707 + 0.707i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4689363714\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4689363714\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.258 - 0.965i)T \) |
good | 3 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.842573308001186823579601671762, −9.353555466183142193950587134566, −8.302280563140820867792263492190, −7.14350181148543171066298050209, −6.55559589055887519138797819771, −5.86230967544072644133222948900, −4.59767241502319932225854862880, −3.53950203010784731419500137033, −2.59268719772469612993072933790, −1.29239733603842157546614137018,
0.55707521654008120960610741365, 2.28140023654852890529242369263, 4.00241638242476319304125393685, 5.16194244002932341430205883884, 5.75731405998724692373969345679, 6.04047599428619131581134227959, 7.10176915062976801175139942790, 8.194655841665327444991199494747, 8.857615415436197382039058564053, 9.655030988029866946042825329271