L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.465 − 0.607i)5-s + (0.707 + 0.707i)8-s + (−0.965 + 0.258i)9-s + (−0.465 + 0.607i)10-s + 2i·13-s + (0.500 − 0.866i)16-s + (0.258 − 0.965i)17-s + (0.499 + 0.866i)18-s + (0.707 + 0.292i)20-s + (0.107 − 0.400i)25-s + (1.93 − 0.517i)26-s + (0.707 − 1.70i)29-s + (−0.965 − 0.258i)32-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.465 − 0.607i)5-s + (0.707 + 0.707i)8-s + (−0.965 + 0.258i)9-s + (−0.465 + 0.607i)10-s + 2i·13-s + (0.500 − 0.866i)16-s + (0.258 − 0.965i)17-s + (0.499 + 0.866i)18-s + (0.707 + 0.292i)20-s + (0.107 − 0.400i)25-s + (1.93 − 0.517i)26-s + (0.707 − 1.70i)29-s + (−0.965 − 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7266444728\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7266444728\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.258 + 0.965i)T \) |
good | 3 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 5 | \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \) |
| 11 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 37 | \( 1 + (-1.46 + 1.12i)T + (0.258 - 0.965i)T^{2} \) |
| 41 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.758 + 0.0999i)T + (0.965 - 0.258i)T^{2} \) |
| 79 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
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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
−8.835960812067333055289537133690, −8.021784841508155512364077114459, −7.34094383381050651405157367914, −6.25777628033056976022327901577, −5.25709719847957892882688235890, −4.42972531757706018779087759008, −3.96430985953028353199041307395, −2.71977948903573184038203509576, −2.03042327918176741754861910213, −0.57743158349372447365651190460,
1.08661392276431142217986290002, 3.00331807577184086874212112397, 3.42124044840710510385749605313, 4.69076839812427809893276012508, 5.49314348326376090580331220784, 6.09576630551531890900616614769, 6.81335732376923022784284275154, 7.73008590383441957500773571061, 8.176041226232431181798090771574, 8.721297099732867453834610963846