L(s) = 1 | + (−0.448 − 0.258i)2-s + (−0.366 − 0.633i)4-s + 0.896i·8-s − 1.93i·11-s + (−0.133 + 0.232i)16-s + (−0.499 + 0.866i)22-s − 1.41i·23-s + 25-s + (−1.22 + 0.707i)29-s + (0.896 − 0.517i)32-s + (0.866 + 1.5i)37-s + (−0.866 − 1.5i)43-s + (−1.22 + 0.707i)44-s + (−0.366 + 0.633i)46-s + (−0.448 − 0.258i)50-s + ⋯ |
L(s) = 1 | + (−0.448 − 0.258i)2-s + (−0.366 − 0.633i)4-s + 0.896i·8-s − 1.93i·11-s + (−0.133 + 0.232i)16-s + (−0.499 + 0.866i)22-s − 1.41i·23-s + 25-s + (−1.22 + 0.707i)29-s + (0.896 − 0.517i)32-s + (0.866 + 1.5i)37-s + (−0.866 − 1.5i)43-s + (−1.22 + 0.707i)44-s + (−0.366 + 0.633i)46-s + (−0.448 − 0.258i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6653653586\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6653653586\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.93iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−8.554646954230867738391888961867, −7.927001410987601618239050162700, −6.69021503946809839000162444174, −6.14349481176473752146328099442, −5.33970604590407859009142441907, −4.70294899942881768737566191399, −3.55198102112787491142878937702, −2.76127541702328020114893236623, −1.54187036197095550472126849503, −0.45984063823424267984120428279,
1.48920090802236468010164958472, 2.59655790297643723403200401274, 3.65846883298553042115121849468, 4.39528805680491112611635197698, 5.05751182772870397688502578275, 6.14606077210635931847613214876, 7.04909885095690567263490293352, 7.54792299048725992068210687139, 7.986818616569090869237486464507, 9.117680006782801880185431651798