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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.216303911\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216303911\) |
\(L(1)\) |
|
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
−10.64940554845876915518082276543, −9.721559713136915931993482560824, −8.945645276797791331027864148445, −8.100283771869481606588448258401, −7.23657877521383497767263340020, −6.31594235116094141169306955317, −4.87568833691894492966308118606, −4.06236654040940025278172943283, −3.36019090097599852242217091277, −1.73880738035066004212755957453,
1.66274092463359388474131334946, 2.81946286797672009368096705447, 3.70082047242043809456201066502, 5.43626017265676189353926315350, 5.97765466754364041880612449728, 7.14117594131401513683028128746, 7.991859654604900860551708282025, 8.874579815573019611835329710387, 9.180718170801788398644967488632, 10.48517391414318431256773648419