L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (0.216 − 0.324i)5-s + (−0.382 − 0.923i)8-s + (0.923 − 0.382i)9-s + (−0.324 + 0.216i)10-s + (1 + i)13-s + i·16-s + (−0.923 − 0.382i)17-s − 18-s + (0.382 − 0.0761i)20-s + (0.324 + 0.783i)25-s + (−0.541 − 1.30i)26-s + (1.08 − 1.63i)29-s + (0.382 − 0.923i)32-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (0.216 − 0.324i)5-s + (−0.382 − 0.923i)8-s + (0.923 − 0.382i)9-s + (−0.324 + 0.216i)10-s + (1 + i)13-s + i·16-s + (−0.923 − 0.382i)17-s − 18-s + (0.382 − 0.0761i)20-s + (0.324 + 0.783i)25-s + (−0.541 − 1.30i)26-s + (1.08 − 1.63i)29-s + (0.382 − 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9796364601\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9796364601\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.923 + 0.382i)T \) |
good | 3 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 97 | \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \) |
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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
−8.871568637981204138915909244831, −8.185539131850127021314696896926, −7.35459439585070526283671362915, −6.55718332838194758969820850564, −6.15229122956769071339167495867, −4.62883011599458187881747224776, −4.07941888161833627716281405986, −2.99750412133669500201882345690, −1.90578955481029503645851528710, −1.06585308055232875366273062727,
1.07898005098471423464073347954, 2.11430303594357086416553983875, 3.11125772861780942345976302348, 4.31107454926894357194241050648, 5.26720712969063210985339010097, 6.09953798749629254573943390255, 6.75583358453089236390328400019, 7.36936508676976103419372329879, 8.260064043326101260887377952568, 8.698473275393610416527533053154