L(s) = 1 | + (−0.900 + 0.433i)3-s + (−0.623 − 0.781i)7-s + (0.623 − 0.781i)9-s + (−0.678 + 0.541i)13-s − 1.24·19-s + (0.900 + 0.433i)21-s + (−0.623 + 0.781i)25-s + (−0.222 + 0.974i)27-s − 0.445·31-s + (−0.0990 − 0.433i)37-s + (0.376 − 0.781i)39-s + (−0.678 + 1.40i)43-s + (−0.222 + 0.974i)49-s + (1.12 − 0.541i)57-s + (−1.90 + 0.433i)61-s + ⋯ |
L(s) = 1 | + (−0.900 + 0.433i)3-s + (−0.623 − 0.781i)7-s + (0.623 − 0.781i)9-s + (−0.678 + 0.541i)13-s − 1.24·19-s + (0.900 + 0.433i)21-s + (−0.623 + 0.781i)25-s + (−0.222 + 0.974i)27-s − 0.445·31-s + (−0.0990 − 0.433i)37-s + (0.376 − 0.781i)39-s + (−0.678 + 1.40i)43-s + (−0.222 + 0.974i)49-s + (1.12 − 0.541i)57-s + (−1.90 + 0.433i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08451115145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08451115145\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.623 + 0.781i)T \) |
good | 5 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 13 | \( 1 + (0.678 - 0.541i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 + 1.24T + T^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (1.90 - 0.433i)T + (0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 - 0.867iT - T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (1.52 + 1.21i)T + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 + 1.94iT - T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 - 1.56iT - T^{2} \) |
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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
−9.540794710224899977160209248507, −9.108521887744258589563839895551, −7.82820442892413459190302692678, −7.07571263718159294108424436437, −6.42839148979578039478152912438, −5.71123465175829811934218338356, −4.63625226205849335249897091574, −4.12919919503096215658839814640, −3.12812750130607533959018301383, −1.62699963464516046973364997464,
0.06115105804545111330799261571, 1.86605317111521047924546743961, 2.76083544350342811163080432453, 4.05402854133169001767344594205, 5.02272679313004864992725825942, 5.74667486340957414375928611331, 6.40099949453583228849668289332, 7.08852769977939005493704477149, 8.004080117020178582472179420768, 8.718578844022016021593887715553