L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 + 0.866i)4-s + (−0.258 − 0.448i)5-s + (−0.258 − 0.965i)6-s + 0.999i·8-s + 1.00i·9-s − 0.517i·10-s + (0.258 − 0.965i)12-s + (−1.22 + 0.707i)13-s + (−0.133 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s + 1.93i·19-s + (0.258 − 0.448i)20-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 + 0.866i)4-s + (−0.258 − 0.448i)5-s + (−0.258 − 0.965i)6-s + 0.999i·8-s + 1.00i·9-s − 0.517i·10-s + (0.258 − 0.965i)12-s + (−1.22 + 0.707i)13-s + (−0.133 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s + 1.93i·19-s + (0.258 − 0.448i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.105070558\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105070558\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.93iT - T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−8.547964202662124504714288854621, −7.977372074878969509952966538066, −7.37686226899432917003102426641, −6.70272717068835276734044368302, −5.84348428060557406341314516022, −5.41240329347475066091063475220, −4.45150693192377754555810129274, −3.88084646101948742872495817432, −2.49482337431768387486689217039, −1.65096595252333402814803175165,
0.49792725279795180731054239591, 2.29569236629904224395054412134, 3.06936611156544679061754938173, 3.93676095084480345233941139564, 4.80342716109298904117059801697, 5.18590578769867613105222848073, 6.14429272205118999563739406814, 6.82633536465333308237746538402, 7.46776965118387026457418347325, 8.716256959358050996044721806960