L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.258 + 0.965i)5-s + (0.5 − 0.866i)7-s + (−0.707 + 0.707i)8-s + 10-s + (0.448 + 0.258i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.258 − 0.965i)20-s + (0.366 − 0.366i)22-s + (−0.866 + 0.499i)25-s + (−0.866 + 0.5i)28-s + 1.93·29-s + (0.866 − 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.258 + 0.965i)5-s + (0.5 − 0.866i)7-s + (−0.707 + 0.707i)8-s + 10-s + (0.448 + 0.258i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.258 − 0.965i)20-s + (0.366 − 0.366i)22-s + (−0.866 + 0.499i)25-s + (−0.866 + 0.5i)28-s + 1.93·29-s + (0.866 − 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.418647850\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418647850\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.258 - 0.965i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 11 | \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 - 1.93T + T^{2} \) |
| 31 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
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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
−9.200876031606048797439481930439, −8.245383348345092408853837144472, −7.46368774638921253951332817347, −6.54740378402180785845623114776, −5.85082888384187495588323170786, −4.65667046599941111823052389951, −4.13028352869219741151580845356, −3.12822044867230113948935167455, −2.29417936912339871320025238626, −1.14382914784148481329963440074,
1.21354967044264094691340569144, 2.68321378210836323913352064999, 3.86916830248358277741343485253, 4.92041274712850289776835967107, 5.14485435626499306822652987479, 6.20098260333993653579199049125, 6.69731143440582654644501230860, 7.948069171947057717129367179930, 8.425266045743047748414688975382, 8.927805939118164253701186198860