L(s) = 1 | + (0.968 − 0.248i)3-s + (−0.809 + 0.587i)4-s + (0.348 + 1.82i)7-s + (0.876 − 0.481i)9-s + (−0.637 + 0.770i)12-s + (−1.62 − 0.645i)13-s + (0.309 − 0.951i)16-s + (0.688 − 0.500i)19-s + (0.791 + 1.68i)21-s + (0.535 − 0.844i)25-s + (0.728 − 0.684i)27-s + (−1.35 − 1.27i)28-s + (0.371 − 0.0469i)31-s + (−0.425 + 0.904i)36-s + (−1.80 − 0.713i)37-s + ⋯ |
L(s) = 1 | + (0.968 − 0.248i)3-s + (−0.809 + 0.587i)4-s + (0.348 + 1.82i)7-s + (0.876 − 0.481i)9-s + (−0.637 + 0.770i)12-s + (−1.62 − 0.645i)13-s + (0.309 − 0.951i)16-s + (0.688 − 0.500i)19-s + (0.791 + 1.68i)21-s + (0.535 − 0.844i)25-s + (0.728 − 0.684i)27-s + (−1.35 − 1.27i)28-s + (0.371 − 0.0469i)31-s + (−0.425 + 0.904i)36-s + (−1.80 − 0.713i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9856608580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9856608580\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.968 + 0.248i)T \) |
| 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 | \( 1 + (-0.348 - 1.82i)T + (-0.929 + 0.368i)T^{2} \) |
| 11 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 13 | \( 1 + (1.62 + 0.645i)T + (0.728 + 0.684i)T^{2} \) |
| 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 | \( 1 + (-0.598 - 0.153i)T + (0.876 + 0.481i)T^{2} \) |
| 67 | \( 1 + (1.11 + 0.614i)T + (0.535 + 0.844i)T^{2} \) |
| 71 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 73 | \( 1 + (-0.303 - 1.58i)T + (-0.929 + 0.368i)T^{2} \) |
| 79 | \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \) |
| 83 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 89 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 97 | \( 1 + (-0.541 - 0.297i)T + (0.535 + 0.844i)T^{2} \) |
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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
−11.83536465702861982416616191814, −10.10755130003660714696545471858, −9.305426866817933584078716102020, −8.685472078829080261567703427031, −7.993182616294991788771077174447, −7.06964267456176411432321712492, −5.45708671223543626538538002387, −4.66825033093199133663121293061, −3.11031893065669380077745523286, −2.37215847792831890311071634457,
1.54448663124299700064814488540, 3.43202387548862539154623412792, 4.43264558870637412839202845055, 5.05578315842562824004733741165, 6.95932935060834329385269967883, 7.54145471134431530847951965860, 8.582750332563762616595703711426, 9.636685688887504771113110556697, 10.07627399886954081158952246426, 10.82999977983433398537784773703