L(s) = 1 | + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (0.382 + 0.923i)8-s + (0.991 + 0.130i)9-s + (1.75 + 0.867i)11-s + (−0.866 − 0.499i)16-s + (−0.866 + 0.499i)18-s + (−1.92 + 0.382i)22-s + (0.0999 − 0.758i)23-s + (0.130 + 0.991i)25-s + (0.324 + 0.216i)29-s + (0.991 − 0.130i)32-s + (0.382 − 0.923i)36-s + (−1.09 + 1.25i)37-s + (−0.923 − 1.38i)43-s + (1.29 − 1.47i)44-s + ⋯ |
L(s) = 1 | + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (0.382 + 0.923i)8-s + (0.991 + 0.130i)9-s + (1.75 + 0.867i)11-s + (−0.866 − 0.499i)16-s + (−0.866 + 0.499i)18-s + (−1.92 + 0.382i)22-s + (0.0999 − 0.758i)23-s + (0.130 + 0.991i)25-s + (0.324 + 0.216i)29-s + (0.991 − 0.130i)32-s + (0.382 − 0.923i)36-s + (−1.09 + 1.25i)37-s + (−0.923 − 1.38i)43-s + (1.29 − 1.47i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.032900276\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.032900276\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.793 - 0.608i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 5 | \( 1 + (-0.130 - 0.991i)T^{2} \) |
| 11 | \( 1 + (-1.75 - 0.867i)T + (0.608 + 0.793i)T^{2} \) |
| 13 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 23 | \( 1 + (-0.0999 + 0.758i)T + (-0.965 - 0.258i)T^{2} \) |
| 29 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.09 - 1.25i)T + (-0.130 - 0.991i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.996 + 0.491i)T + (0.608 + 0.793i)T^{2} \) |
| 59 | \( 1 + (-0.793 + 0.608i)T^{2} \) |
| 61 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 67 | \( 1 + (-1.10 - 0.0726i)T + (0.991 + 0.130i)T^{2} \) |
| 71 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 79 | \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \) |
| 83 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 89 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 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
−8.958519737601065897875309407557, −8.324218888292417296323983243708, −7.30165686042782386776045621775, −6.84445652529083384368940021533, −6.37368857846381630279499649311, −5.14978476815146941142951860885, −4.52389011361527329914841022311, −3.53695586525919365266324033287, −1.94956064409716471770684938553, −1.29059501701989550919589749231,
1.02547952238984887346606355049, 1.83275958220814311544877204449, 3.16116615764887783913117558632, 3.86832812457255277242650221050, 4.55452847674206562376899478383, 5.97879199940829869912812148805, 6.71755087713034775103929958747, 7.27232099843921505544415273195, 8.269949833116581545756070530270, 8.798815282569391194007324267556