L(s) = 1 | + (0.809 − 0.587i)4-s + (0.723 − 0.137i)7-s + (−0.354 + 0.895i)13-s + (0.309 − 0.951i)16-s + (0.688 − 0.500i)19-s + (−0.535 + 0.844i)25-s + (0.503 − 0.536i)28-s + (0.371 − 0.0469i)31-s + (−1.80 − 0.713i)37-s + (0.0235 − 0.123i)43-s + (−0.425 + 0.168i)49-s + (0.239 + 0.933i)52-s + (0.473 − 1.84i)61-s + (−0.309 − 0.951i)64-s + (−0.742 + 1.35i)67-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)4-s + (0.723 − 0.137i)7-s + (−0.354 + 0.895i)13-s + (0.309 − 0.951i)16-s + (0.688 − 0.500i)19-s + (−0.535 + 0.844i)25-s + (0.503 − 0.536i)28-s + (0.371 − 0.0469i)31-s + (−1.80 − 0.713i)37-s + (0.0235 − 0.123i)43-s + (−0.425 + 0.168i)49-s + (0.239 + 0.933i)52-s + (0.473 − 1.84i)61-s + (−0.309 − 0.951i)64-s + (−0.742 + 1.35i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.381667885\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381667885\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 151 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (0.535 - 0.844i)T^{2} \) |
| 7 | \( 1 + (-0.723 + 0.137i)T + (0.929 - 0.368i)T^{2} \) |
| 11 | \( 1 + (0.876 + 0.481i)T^{2} \) |
| 13 | \( 1 + (0.354 - 0.895i)T + (-0.728 - 0.684i)T^{2} \) |
| 17 | \( 1 + (-0.929 - 0.368i)T^{2} \) |
| 19 | \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.992 + 0.125i)T^{2} \) |
| 31 | \( 1 + (-0.371 + 0.0469i)T + (0.968 - 0.248i)T^{2} \) |
| 37 | \( 1 + (1.80 + 0.713i)T + (0.728 + 0.684i)T^{2} \) |
| 41 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 43 | \( 1 + (-0.0235 + 0.123i)T + (-0.929 - 0.368i)T^{2} \) |
| 47 | \( 1 + (-0.425 + 0.904i)T^{2} \) |
| 53 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.473 + 1.84i)T + (-0.876 - 0.481i)T^{2} \) |
| 67 | \( 1 + (0.742 - 1.35i)T + (-0.535 - 0.844i)T^{2} \) |
| 71 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 73 | \( 1 + (-1.15 + 0.220i)T + (0.929 - 0.368i)T^{2} \) |
| 79 | \( 1 + (0.211 - 1.67i)T + (-0.968 - 0.248i)T^{2} \) |
| 83 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 89 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 97 | \( 1 + (-0.541 - 0.297i)T + (0.535 + 0.844i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.752617857214869521927900949928, −9.079052976060389532403331164909, −7.980900166285651382065924083528, −7.19957570575898251393078015624, −6.60524142762001371117034880175, −5.48692973957101858577438739975, −4.88157594279102994258593751292, −3.63966612947502734096714829006, −2.35530393144434659934684250561, −1.43746327434114551256849676678,
1.64029634497934845396570303070, 2.74377732317314072203470304812, 3.64682365223368983205376484356, 4.83926497434524351426989615999, 5.73134134451491571684754020180, 6.65077620084797281601690396801, 7.59451461332997490039947452846, 8.056911095002830646349850278491, 8.839229853166184935102554090255, 10.09925777319010473264232834151