L(s) = 1 | + (0.793 − 0.608i)5-s + (0.866 + 0.5i)9-s + (−0.541 + 0.541i)13-s + (0.478 − 1.78i)17-s + (0.258 − 0.965i)25-s − 1.41i·29-s + (0.517 + 1.93i)37-s − 0.765i·41-s + (0.991 − 0.130i)45-s + (−0.366 + 1.36i)53-s + (1.60 + 0.923i)61-s + (−0.0999 + 0.758i)65-s + (−1.78 − 0.478i)73-s + (0.499 + 0.866i)81-s + (−0.707 − 1.70i)85-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)5-s + (0.866 + 0.5i)9-s + (−0.541 + 0.541i)13-s + (0.478 − 1.78i)17-s + (0.258 − 0.965i)25-s − 1.41i·29-s + (0.517 + 1.93i)37-s − 0.765i·41-s + (0.991 − 0.130i)45-s + (−0.366 + 1.36i)53-s + (1.60 + 0.923i)61-s + (−0.0999 + 0.758i)65-s + (−1.78 − 0.478i)73-s + (0.499 + 0.866i)81-s + (−0.707 − 1.70i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.608297393\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.608297393\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.793 + 0.608i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 17 | \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + 0.765iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
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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
−8.696358143695240585103962442378, −7.74065845048154658784118847085, −7.20656476810111780840406174094, −6.37339735985710400985152353964, −5.51523762732652894597236929835, −4.75956931242118190008385768078, −4.32286267813383801777840362948, −2.89191777140031939999129033948, −2.11460060210026892770636273291, −1.04984633787087071444886035687,
1.34256831903997319395807278827, 2.21552425600660151434293592183, 3.32517142781782395039134914499, 3.96608629379632026969066007405, 5.09911076583759005315648494763, 5.80449049587021431529521091221, 6.51830231797661923038663009140, 7.15266152344893349109050268087, 7.897243252996568563497430707466, 8.756400538122844241157240636970