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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.751829171\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.751829171\) |
\(L(1)\) |
|
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
−9.213382902528685623944695872790, −8.319405044629797066379231447373, −7.66866800146716269980695574243, −7.27677687528566632974000548620, −6.02223946913005617378765071605, −4.92354595125333116942223091019, −4.41905042605301835567597193727, −3.40419593127907856301500176221, −2.58905056271359823930045397405, −1.34581639210441885381041289027,
1.48657834125496618821922384319, 2.35756799934450588974114719000, 3.23809845493727868852194853221, 4.27873992943529624391779349223, 5.22248942342986568734996506745, 6.04966554863206445841782250330, 7.15792042265626909352049226642, 7.59785251265433126890598774089, 8.386158403263398536375017221519, 9.220822426325422108086155984248