| L(s) = 1 | − 2.53·2-s + 3-s + 4.41·4-s + 0.879·5-s − 2.53·6-s − 4.41·7-s − 6.10·8-s + 9-s − 2.22·10-s + 3.71·11-s + 4.41·12-s − 3.12·13-s + 11.1·14-s + 0.879·15-s + 6.63·16-s − 2.53·18-s − 2.04·19-s + 3.87·20-s − 4.41·21-s − 9.41·22-s + 0.162·23-s − 6.10·24-s − 4.22·25-s + 7.90·26-s + 27-s − 19.4·28-s − 8.33·29-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.20·4-s + 0.393·5-s − 1.03·6-s − 1.66·7-s − 2.15·8-s + 0.333·9-s − 0.704·10-s + 1.12·11-s + 1.27·12-s − 0.865·13-s + 2.98·14-s + 0.227·15-s + 1.65·16-s − 0.596·18-s − 0.468·19-s + 0.867·20-s − 0.962·21-s − 2.00·22-s + 0.0338·23-s − 1.24·24-s − 0.845·25-s + 1.54·26-s + 0.192·27-s − 3.67·28-s − 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 19 | \( 1 + 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.162T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 + 2.50T + 61T^{2} \) |
| 67 | \( 1 + 4.61T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 - 2.77T + 73T^{2} \) |
| 79 | \( 1 - 3.45T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 4.30T + 97T^{2} \) |
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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
−9.589099502443781957623519488124, −9.218769701958379506137607487838, −8.283614450900851061209976473084, −7.24432180227590712434881493659, −6.71866892552215793266959468263, −5.87239056613856141217095348740, −3.87343020904068873513726760781, −2.77298242424866877592942047125, −1.71730405141237852819488748932, 0,
1.71730405141237852819488748932, 2.77298242424866877592942047125, 3.87343020904068873513726760781, 5.87239056613856141217095348740, 6.71866892552215793266959468263, 7.24432180227590712434881493659, 8.283614450900851061209976473084, 9.218769701958379506137607487838, 9.589099502443781957623519488124
