L(s) = 1 | + (−0.222 − 0.974i)4-s + (−0.988 − 0.149i)7-s + (0.603 + 1.53i)13-s + (−0.900 + 0.433i)16-s + (−0.623 + 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.0747 + 0.997i)28-s − 1.46·31-s + (1.07 + 0.997i)37-s + (−1.63 − 1.11i)43-s + (0.955 + 0.294i)49-s + (1.36 − 0.930i)52-s + (−0.425 + 1.86i)61-s + (0.623 + 0.781i)64-s + 0.149·67-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)4-s + (−0.988 − 0.149i)7-s + (0.603 + 1.53i)13-s + (−0.900 + 0.433i)16-s + (−0.623 + 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.0747 + 0.997i)28-s − 1.46·31-s + (1.07 + 0.997i)37-s + (−1.63 − 1.11i)43-s + (0.955 + 0.294i)49-s + (1.36 − 0.930i)52-s + (−0.425 + 1.86i)61-s + (0.623 + 0.781i)64-s + 0.149·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5620410256\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5620410256\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.988 + 0.149i)T \) |
good | 2 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 11 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 13 | \( 1 + (-0.603 - 1.53i)T + (-0.733 + 0.680i)T^{2} \) |
| 17 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 19 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 31 | \( 1 + 1.46T + T^{2} \) |
| 37 | \( 1 + (-1.07 - 0.997i)T + (0.0747 + 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 43 | \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \) |
| 47 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 61 | \( 1 + (0.425 - 1.86i)T + (-0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 - 0.149T + T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.162 - 0.414i)T + (-0.733 - 0.680i)T^{2} \) |
| 79 | \( 1 - 1.91T + T^{2} \) |
| 83 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 89 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 97 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.075624991037188656060193684401, −8.193248734426534513151062544126, −7.14226788196182248284510360431, −6.43975186061662350284991720243, −6.03310816311104336376516204949, −5.17262212371983489505503136958, −4.10733271182417323973062036497, −3.68012415022423422337170019602, −2.23207445564970490958627529114, −1.41443713471656574775465933126,
0.31066957903571360051885695811, 2.21028911304560283433176797995, 3.16518491369765702970892892021, 3.59737381482700219018640810837, 4.57720209118555621285538523180, 5.54181905417636874466292087098, 6.29043860054004513289145193821, 7.03809897716107182147173465334, 7.86630748125910185536683364982, 8.318423813431723816365475003411