L(s) = 1 | + (0.707 + 0.707i)3-s + 1.00i·9-s + (1.41 + 1.41i)19-s − i·25-s + (−0.707 + 0.707i)27-s + (−1.41 + 1.41i)43-s + 49-s + 2.00i·57-s + (1.41 + 1.41i)67-s − 2i·73-s + (0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + 1.00i·9-s + (1.41 + 1.41i)19-s − i·25-s + (−0.707 + 0.707i)27-s + (−1.41 + 1.41i)43-s + 49-s + 2.00i·57-s + (1.41 + 1.41i)67-s − 2i·73-s + (0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.604183966\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.604183966\) |
\(L(1)\) |
|
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 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.073221786252299923806390063414, −8.200111280201559902131618305349, −7.83501986510574242241607440647, −6.87203775467530392545073529481, −5.84708554138472942478538196571, −5.12709580101326776447169180230, −4.24377471230616338001811570280, −3.48088156479517302531132840367, −2.68939251459845200889645947595, −1.53350743937574894770540309698,
0.994218263864976680754827617339, 2.15055032143309626007799534134, 3.06496364151215791319726696955, 3.77662218771910359513202118686, 4.98295731119253467662722568788, 5.70008998277433213533701522031, 6.89726913440852268164912887252, 7.10191604597371451993263308553, 8.001015597769676503500658238449, 8.723918046626110272180542179297