L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + 1.00·10-s + (−0.707 + 0.707i)11-s + i·13-s − 1.00·16-s + (−0.707 − 0.707i)17-s + (−0.707 + 0.707i)20-s − 1.00i·22-s + (−0.707 + 0.707i)23-s + 1.00i·25-s + (−0.707 − 0.707i)26-s + (0.707 + 0.707i)29-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + 1.00·10-s + (−0.707 + 0.707i)11-s + i·13-s − 1.00·16-s + (−0.707 − 0.707i)17-s + (−0.707 + 0.707i)20-s − 1.00i·22-s + (−0.707 + 0.707i)23-s + 1.00i·25-s + (−0.707 − 0.707i)26-s + (0.707 + 0.707i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4226363575\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4226363575\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - iT - T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 - 2iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−9.343830888583587316326771434233, −8.801904447102993677529781969046, −7.79276490462752605072063982899, −7.51055036073531728977647840050, −6.57756054427567159612012094881, −5.67446966181710748852418363260, −4.67306372519341234469690156285, −4.29337718196958449001127094337, −2.57723271062309493000018427452, −1.35529478392443849270202305131,
0.38826245540949152131986365804, 2.17243039326751140882828853663, 3.03901239264438875328493908012, 3.75032821598482199289521544378, 4.73286837016023124939932928017, 6.02080508406673625062044473748, 6.84402281924162255184241859533, 7.76627270096701959519447230154, 8.257589023780912760526907398741, 8.815169709644298952323206884349