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 + (0.500 + 0.866i)10-s + (0.500 + 0.866i)16-s − 1.93i·17-s + (−1.36 + 1.36i)19-s + (0.258 + 0.965i)20-s − 0.517i·23-s + 1.00i·25-s − i·31-s + (0.258 + 0.965i)32-s + (0.499 − 1.86i)34-s + (−1.67 + 0.965i)38-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 + (0.500 + 0.866i)10-s + (0.500 + 0.866i)16-s − 1.93i·17-s + (−1.36 + 1.36i)19-s + (0.258 + 0.965i)20-s − 0.517i·23-s + 1.00i·25-s − i·31-s + (0.258 + 0.965i)32-s + (0.499 − 1.86i)34-s + (−1.67 + 0.965i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.376580992\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.376580992\) |
\(L(1)\) |
|
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 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + 1.93iT - T^{2} \) |
| 19 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 23 | \( 1 + 0.517iT - T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.73iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 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
−9.532924772942023395029072563590, −8.399427102538680705137678398780, −7.61336131539428460921546445328, −6.78846927650153977302022162897, −6.25196661599153502436517019205, −5.45340235435752239153568091888, −4.62700780862877834471259954330, −3.62966650219597926107340285045, −2.70261032758951081776952933469, −1.92755943851789876013732800476,
1.44752784203497121782941210547, 2.26401617166765331202112614429, 3.43974815952142843034827813110, 4.44794710118329365147222029407, 4.98042595590914922364374559324, 6.09498491834887201252596059624, 6.33476454290578036289107969330, 7.46336232726100468283310604403, 8.515634673683658597499802206179, 9.101294826936323371816291297200