L(s) = 1 | + (−0.623 + 0.781i)3-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)9-s + (1.52 + 0.347i)13-s − 0.445·19-s + (−0.623 − 0.781i)21-s + (0.222 + 0.974i)25-s + (0.900 + 0.433i)27-s + 1.80·31-s + (−1.62 + 0.781i)37-s + (−1.22 + 0.974i)39-s + (−1.52 + 1.21i)43-s + (−0.900 − 0.433i)49-s + (0.277 − 0.347i)57-s + (−0.376 − 0.781i)61-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)3-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)9-s + (1.52 + 0.347i)13-s − 0.445·19-s + (−0.623 − 0.781i)21-s + (0.222 + 0.974i)25-s + (0.900 + 0.433i)27-s + 1.80·31-s + (−1.62 + 0.781i)37-s + (−1.22 + 0.974i)39-s + (−1.52 + 1.21i)43-s + (−0.900 − 0.433i)49-s + (0.277 − 0.347i)57-s + (−0.376 − 0.781i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8998503948\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8998503948\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.623 - 0.781i)T \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
good | 5 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (1.52 - 1.21i)T + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 - 1.56iT - T^{2} \) |
| 71 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.846 + 0.193i)T + (0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + 0.867iT - T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 - 1.94iT - T^{2} \) |
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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
−9.366176225525895904858723636490, −8.716646217121041472310414546473, −8.175030017413761137175756752368, −6.61496354846829600230477167721, −6.35873092574648860767359434058, −5.42332752677594433292090150018, −4.74288479931963217122658685460, −3.70673848974345101078448718270, −2.97727333996728002763040828790, −1.46523429046331886913914401604,
0.71995210668647087997647662698, 1.82493380605408697258664078838, 3.17488670903551823327707411186, 4.12989585224044429505395681893, 5.05879144261738747321381003459, 6.06701711214470181316273645725, 6.58734948654215758674797991180, 7.24951787682004774617022100795, 8.262765729472943960745617735188, 8.571746181339135024403365704993