L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.258 + 0.965i)3-s + (−0.866 + 0.499i)4-s + (−0.258 + 0.965i)5-s + 6-s + (1.22 − 0.707i)7-s + (0.707 + 0.707i)8-s + 10-s + (1 + i)11-s + (−0.258 − 0.965i)12-s + (−0.965 − 0.258i)13-s + (−1 − 0.999i)14-s + (−0.866 − 0.499i)15-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.258 + 0.965i)3-s + (−0.866 + 0.499i)4-s + (−0.258 + 0.965i)5-s + 6-s + (1.22 − 0.707i)7-s + (0.707 + 0.707i)8-s + 10-s + (1 + i)11-s + (−0.258 − 0.965i)12-s + (−0.965 − 0.258i)13-s + (−1 − 0.999i)14-s + (−0.866 − 0.499i)15-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.130418877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130418877\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 + (0.258 - 0.965i)T \) |
| 37 | \( 1 + (-0.258 + 0.965i)T \) |
good | 3 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1 - i)T + iT^{2} \) |
| 13 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−9.569781173557735340725772566765, −8.150269261511203350857755786335, −7.65590269687315847357190838189, −7.05348966982074573650212758716, −5.61174981850091675418937747204, −4.73449608362032427325607188196, −4.17379853963877281191273752067, −3.59068023321267912841357674426, −2.39861263234242801513711742782, −1.34080451865814446633015855327,
1.06507529529319015343154828045, 1.51991581921127899845503661196, 3.36587358769021412661566071832, 4.59096543928612690843200636006, 5.29305606788475678700855641933, 5.69602109969851110766889152043, 6.73862711055188644142659831229, 7.60092282683091641980271228995, 7.81559510394452640726305134090, 8.708227743636856422010284231456