L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)5-s − i·7-s + (0.707 − 0.707i)8-s + 1.00i·10-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)5-s − i·7-s + (0.707 − 0.707i)8-s + 1.00i·10-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 1.41i·17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + 1.41i·26-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5100651145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5100651145\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−11.11294518251102064731607168112, −10.01909921113000074077632819049, −9.547463512100229922306465248694, −8.353983810850964368888156806829, −7.58874656069789216065624730099, −6.90576961626624250002864950432, −4.85370867393979454556579685243, −4.14131322971884695454873142521, −2.80006230187750664070888616745, −0.906237298384295704270118633130,
2.11170076498663153757087653620, 3.78348720928138463962681262861, 5.20212322128511737537263313873, 6.34078008816720197084693177101, 6.95329677180448468545439805557, 8.122050644798931139573399530374, 8.768093824882733956030420539145, 9.686737008859661650087056706372, 10.67237083024378734677860303803, 11.58301526492345170214753952651