L(s) = 1 | + 1.41·2-s + 2.00·4-s + (−0.707 − 0.707i)5-s − 4.82i·7-s + 2.82·8-s + (−1.00 − 1.00i)10-s + (−1.41 − 1.41i)11-s + (−0.585 + 0.585i)13-s − 6.82i·14-s + 4.00·16-s − 5.41·17-s + (3.82 − 3.82i)19-s + (−1.41 − 1.41i)20-s + (−2.00 − 2.00i)22-s + 5.41i·23-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 1.00·4-s + (−0.316 − 0.316i)5-s − 1.82i·7-s + 1.00·8-s + (−0.316 − 0.316i)10-s + (−0.426 − 0.426i)11-s + (−0.162 + 0.162i)13-s − 1.82i·14-s + 1.00·16-s − 1.31·17-s + (0.878 − 0.878i)19-s + (−0.316 − 0.316i)20-s + (−0.426 − 0.426i)22-s + 1.12i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16312 - 1.44535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16312 - 1.44535i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + 4.82iT - 7T^{2} \) |
| 11 | \( 1 + (1.41 + 1.41i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.585 - 0.585i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + (-3.82 + 3.82i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.41iT - 23T^{2} \) |
| 29 | \( 1 + (-0.585 + 0.585i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + (-4.58 - 4.58i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.82iT - 41T^{2} \) |
| 43 | \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.41 - 7.41i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9.48 + 9.48i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.65 - 7.65i)T - 67iT^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 - 3.17iT - 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + (-3.07 + 3.07i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−10.54144164377034142358467421763, −9.600211087517009599667182154763, −8.254468957406241669483772980213, −7.29501512797475085966123544662, −6.89723847577198610426877296343, −5.59652841924609334790825675525, −4.52806857079287912577582144176, −3.97680140801205382686129819548, −2.78377945845621131275261528742, −1.02231951697759098449684668105,
2.21061334529709178231934295255, 2.82357140665363153117028514468, 4.20807900385427558275043333716, 5.20548752632221928563298398664, 5.96143437341199722836198636853, 6.82724929963524474710323231495, 7.896624193449233314849871874746, 8.737066591210459943392487870405, 9.842523997209935562367725258777, 10.81756783022908837450876434727