L(s) = 1 | + (−0.991 + 0.130i)5-s + (−0.866 + 0.5i)9-s + (1.30 − 1.30i)13-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)25-s + 1.41i·29-s + (1.93 + 0.517i)37-s − 1.84i·41-s + (0.793 − 0.608i)45-s + (1.36 − 0.366i)53-s + (0.662 − 0.382i)61-s + (−1.12 + 1.46i)65-s + (0.198 + 0.739i)73-s + (0.499 − 0.866i)81-s + (0.707 − 0.292i)85-s + ⋯ |
L(s) = 1 | + (−0.991 + 0.130i)5-s + (−0.866 + 0.5i)9-s + (1.30 − 1.30i)13-s + (−0.739 + 0.198i)17-s + (0.965 − 0.258i)25-s + 1.41i·29-s + (1.93 + 0.517i)37-s − 1.84i·41-s + (0.793 − 0.608i)45-s + (1.36 − 0.366i)53-s + (0.662 − 0.382i)61-s + (−1.12 + 1.46i)65-s + (0.198 + 0.739i)73-s + (0.499 − 0.866i)81-s + (0.707 − 0.292i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9923478402\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9923478402\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.991 - 0.130i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 17 | \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + 1.84iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
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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
−8.564851719676318682615960673103, −8.012096357538373957010166915185, −7.31444955242770765396265256093, −6.41553970235025093070722201913, −5.65576990759260960505105057192, −4.92631672733976368179367818382, −3.89186790067308571823766271824, −3.28506403691536844211228298338, −2.38192765883766882814382479588, −0.822077296150696401396145358258,
0.869743149276443668558106479813, 2.30324857058388436763044949293, 3.31046664378100697153528092199, 4.13930330983861298642817261116, 4.59242978707423446848287961183, 5.94100884482617054555941185440, 6.33191037995147901494297695319, 7.21051963751599658912589412151, 8.040407937976999108385496979436, 8.660884479677266027421476830902