| L(s) = 1 | + 0.375·2-s + 3-s − 1.85·4-s − 2.43·5-s + 0.375·6-s + 3.04·7-s − 1.44·8-s + 9-s − 0.914·10-s + 5.98·11-s − 1.85·12-s + 1.84·13-s + 1.14·14-s − 2.43·15-s + 3.17·16-s − 17-s + 0.375·18-s + 1.50·19-s + 4.52·20-s + 3.04·21-s + 2.24·22-s − 0.789·23-s − 1.44·24-s + 0.934·25-s + 0.692·26-s + 27-s − 5.66·28-s + ⋯ |
| L(s) = 1 | + 0.265·2-s + 0.577·3-s − 0.929·4-s − 1.08·5-s + 0.153·6-s + 1.15·7-s − 0.512·8-s + 0.333·9-s − 0.289·10-s + 1.80·11-s − 0.536·12-s + 0.511·13-s + 0.306·14-s − 0.628·15-s + 0.793·16-s − 0.242·17-s + 0.0884·18-s + 0.346·19-s + 1.01·20-s + 0.665·21-s + 0.479·22-s − 0.164·23-s − 0.295·24-s + 0.186·25-s + 0.135·26-s + 0.192·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.233939323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.233939323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 0.375T + 2T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 - 5.98T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 + 0.789T + 23T^{2} \) |
| 29 | \( 1 + 4.72T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 4.05T + 53T^{2} \) |
| 59 | \( 1 + 0.469T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 7.72T + 89T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
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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
−8.583634224149506919817943174203, −7.71251688678672669855984941452, −7.31121660500454501938304579076, −6.15169651492332422880172880402, −5.29873681223037242970811864005, −4.26273053071244120637852908999, −4.03426262014344309342689722714, −3.38105544781702862856202450446, −1.86535513868455895634830799111, −0.856886794034160018539716482302,
0.856886794034160018539716482302, 1.86535513868455895634830799111, 3.38105544781702862856202450446, 4.03426262014344309342689722714, 4.26273053071244120637852908999, 5.29873681223037242970811864005, 6.15169651492332422880172880402, 7.31121660500454501938304579076, 7.71251688678672669855984941452, 8.583634224149506919817943174203
