L(s) = 1 | + (−1.36 − 0.988i)3-s + (−0.809 + 0.587i)4-s + (−0.404 + 1.24i)5-s + (0.565 + 1.74i)9-s + 1.68·12-s + (1.78 − 1.29i)15-s + (0.309 − 0.951i)16-s + (−0.404 − 1.24i)20-s − 1.91·23-s + (−0.578 − 0.420i)25-s + (0.431 − 1.32i)27-s + (−0.592 − 1.82i)31-s + (−1.48 − 1.07i)36-s + (−0.672 + 0.488i)37-s − 2.39·45-s + ⋯ |
L(s) = 1 | + (−1.36 − 0.988i)3-s + (−0.809 + 0.587i)4-s + (−0.404 + 1.24i)5-s + (0.565 + 1.74i)9-s + 1.68·12-s + (1.78 − 1.29i)15-s + (0.309 − 0.951i)16-s + (−0.404 − 1.24i)20-s − 1.91·23-s + (−0.578 − 0.420i)25-s + (0.431 − 1.32i)27-s + (−0.592 − 1.82i)31-s + (−1.48 − 1.07i)36-s + (−0.672 + 0.488i)37-s − 2.39·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08468834112\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08468834112\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.404 - 1.24i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + 1.91T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.592 + 1.82i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.672 + 0.488i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.30T + T^{2} \) |
| 71 | \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + 0.284T + T^{2} \) |
| 97 | \( 1 + (0.592 + 1.82i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−9.723360119786147504650262256043, −8.361596304660442579447016430722, −7.65283907611406537729419571192, −7.08499761351543973471650429882, −6.23767982440381064270892361462, −5.52620930853638338400892866584, −4.37920296286079656381582381364, −3.43278416229258840300236509988, −2.06548028031878964011245206911, −0.094203593162691583514429885827,
1.31359029467698936436623054699, 3.79965436967675944039204753310, 4.40050957636114654562732970058, 5.10665903087951618533043727291, 5.62703530897772423036995458997, 6.47897599686832003470152907076, 7.923995237524650304546313950565, 8.840913048242415624967441679595, 9.374942441158280503374500295154, 10.25132548616641103829530560870