L(s) = 1 | + (−0.222 − 0.974i)2-s + (0.777 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.777 − 0.974i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (−0.123 + 0.541i)9-s + (0.0990 + 0.433i)11-s + (−1.12 − 0.541i)12-s + (0.0990 + 0.433i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + 0.554·18-s + (−0.277 + 1.21i)21-s + (0.400 − 0.193i)22-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (0.777 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.777 − 0.974i)6-s + (0.623 + 0.781i)7-s + (0.623 + 0.781i)8-s + (−0.123 + 0.541i)9-s + (0.0990 + 0.433i)11-s + (−1.12 − 0.541i)12-s + (0.0990 + 0.433i)13-s + (0.623 − 0.781i)14-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + 0.554·18-s + (−0.277 + 1.21i)21-s + (0.400 − 0.193i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.415068823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.415068823\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
| 17 | \( 1 + (0.900 + 0.433i)T \) |
good | 3 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 61 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.068828447965179267013278841147, −8.520954206152058642092261634863, −7.80405078711136000542316992611, −6.74578121952291483845094408159, −5.49912617461983661980282345266, −4.59501675266173305164309397265, −4.28736852378233598901268534655, −3.16384483852969260352581428576, −2.54178433631714906095271459887, −1.56155869353784588912339738545,
0.906870331352341292476389969740, 1.92063486695949818513799242389, 3.16864521892414047524088377917, 4.25594542847401194036956686213, 4.91679078249261271172406145374, 6.06284910266007748167306223539, 6.66054513027771580912661659015, 7.35896781344060731639351647068, 8.011981819867755251076319122526, 8.396255741131433372408236078824