L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.41i·7-s − 1.00i·9-s + (1 − i)13-s + (1.00 + 1.00i)21-s + i·25-s + (0.707 + 0.707i)27-s + 1.41·31-s + (−1 − i)37-s + 1.41i·39-s − 1.00·49-s + (−1 + i)61-s − 1.41·63-s + (1.41 − 1.41i)67-s + (−0.707 − 0.707i)75-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.41i·7-s − 1.00i·9-s + (1 − i)13-s + (1.00 + 1.00i)21-s + i·25-s + (0.707 + 0.707i)27-s + 1.41·31-s + (−1 − i)37-s + 1.41i·39-s − 1.00·49-s + (−1 + i)61-s − 1.41·63-s + (1.41 − 1.41i)67-s + (−0.707 − 0.707i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7803578202\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7803578202\) |
\(L(1)\) |
|
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 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.59135513172079438255753109550, −9.901708357798275185927603360479, −8.893647310593637398508269619198, −7.84516857620233318507649007436, −6.91575487484178863634416985091, −6.00712842047157907449913149768, −5.05874528787727002927089794351, −4.03831900037673161004832773549, −3.31422842149176062958162581066, −1.02178174975497537173849244892,
1.63729939982513489239532469311, 2.76129092971241327325370869272, 4.40429449059889297546425732176, 5.43722649342321044303429795478, 6.25480779584041687786407899601, 6.81527305640456196656582093577, 8.205118052140293699392076958754, 8.644065087000937561093899440474, 9.744109160255819942570778090151, 10.76131868257603734583484341556