L(s) = 1 | + (0.955 + 0.294i)4-s + (0.623 − 0.781i)7-s + (0.0931 − 0.116i)13-s + (0.826 + 0.563i)16-s + (−0.623 − 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.826 − 0.563i)28-s + (−0.955 + 1.65i)31-s + (1.82 − 0.563i)37-s + (−0.658 + 0.317i)43-s + (−0.222 − 0.974i)49-s + (0.123 − 0.0841i)52-s + (−1.40 + 0.432i)61-s + (0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯ |
L(s) = 1 | + (0.955 + 0.294i)4-s + (0.623 − 0.781i)7-s + (0.0931 − 0.116i)13-s + (0.826 + 0.563i)16-s + (−0.623 − 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.826 − 0.563i)28-s + (−0.955 + 1.65i)31-s + (1.82 − 0.563i)37-s + (−0.658 + 0.317i)43-s + (−0.222 − 0.974i)49-s + (0.123 − 0.0841i)52-s + (−1.40 + 0.432i)61-s + (0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.401849417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.401849417\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
good | 2 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 5 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 11 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 13 | \( 1 + (-0.0931 + 0.116i)T + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 19 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 61 | \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.440 + 0.0663i)T + (0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 97 | \( 1 - 0.730T + T^{2} \) |
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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
−9.964484187881104830187027019787, −8.898858729439210072619014515637, −8.025636077429588630974360727520, −7.35432147502833390700002261100, −6.69972499027653734875214174790, −5.74476523499981902188562181099, −4.64453880334471008996235610463, −3.70429982610737716866988015480, −2.60375470571009684141867570961, −1.46327776776313229163270577211,
1.69263510353415156345136098572, 2.43189292073389740998427817482, 3.71600699375093988717230750029, 4.88027351676927085870128965539, 6.00562923699590937731487376015, 6.20525788248326433027249029302, 7.66596976799529336451474911651, 7.940643047489370963387158194880, 9.107732001437310395481740746971, 9.870968251957206685687320060444