L(s) = 1 | + (−0.965 − 0.258i)3-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)11-s + (−0.5 − 0.866i)13-s + i·17-s + 1.41·19-s + (−0.258 + 0.965i)23-s + (0.866 + 0.5i)25-s + (−0.707 − 0.707i)27-s + (1.36 − 0.366i)29-s + (−0.965 − 0.258i)31-s + 33-s + (0.258 + 0.965i)39-s + (0.707 − 1.22i)43-s + (1.22 + 0.707i)47-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)3-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)11-s + (−0.5 − 0.866i)13-s + i·17-s + 1.41·19-s + (−0.258 + 0.965i)23-s + (0.866 + 0.5i)25-s + (−0.707 − 0.707i)27-s + (1.36 − 0.366i)29-s + (−0.965 − 0.258i)31-s + 33-s + (0.258 + 0.965i)39-s + (0.707 − 1.22i)43-s + (1.22 + 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8128956385\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8128956385\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 17 | \( 1 - iT \) |
good | 5 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 83 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T^{2} \) |
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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
−9.285343174420618359491951760232, −8.132920121995606832710786398258, −7.53105822977160166683313481130, −6.96536348569267888209071552744, −5.74941175843886985802562956542, −5.46457217265679942613953614621, −4.60469754687277721744096578527, −3.45754473282869092802615225927, −2.33758309272302245696104912706, −1.02285966557474855996019303882,
0.802871719656149056769357132621, 2.38941739711614128601662096520, 3.41245154157431612886967947151, 4.75505436204266786402187501887, 4.94319564592731051683698968903, 5.94370664835513427444457732767, 6.82494133990361003265718649957, 7.34405879029662679433337117771, 8.337831814844467370195083288319, 9.304471787642970663066894374755