L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.133 − 0.5i)13-s + (0.500 + 0.866i)16-s + (1.22 − 1.22i)17-s + (−0.965 + 0.258i)20-s − 1.00i·25-s + 0.517i·26-s + (0.965 + 1.67i)29-s + (−0.258 − 0.965i)32-s + (−1.49 + 0.866i)34-s + (1.36 + 1.36i)37-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.133 − 0.5i)13-s + (0.500 + 0.866i)16-s + (1.22 − 1.22i)17-s + (−0.965 + 0.258i)20-s − 1.00i·25-s + 0.517i·26-s + (0.965 + 1.67i)29-s + (−0.258 − 0.965i)32-s + (−1.49 + 0.866i)34-s + (1.36 + 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6552393192\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6552393192\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 41 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + 0.517T + T^{2} \) |
| 97 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.865539740911710090950233399608, −8.740931695519330036621961647416, −8.029315778814311538602630448676, −7.36394626364263331192908922181, −6.76971133520368649514905545141, −5.72757831753283064300644366282, −4.48445875261144099137359628256, −3.15791385918476721056430636178, −2.81585811906944623297504170800, −1.06452112433471316788125502474,
0.935509401911315353931934281498, 2.22045349369652357799691041359, 3.61472686744484970078330187291, 4.59334968228511927275149808637, 5.73458277768552386263427677649, 6.36599951389323460422838151896, 7.67065713863094922282912883657, 7.82792784785084937365259649632, 8.743243349929217249880202121361, 9.443249454301124347836327751155