| L(s) = 1 | + 3.55·2-s + 4.62·4-s + 4.70·5-s − 31.4·7-s − 11.9·8-s + 16.7·10-s + 25.9·11-s + 44.2·13-s − 111.·14-s − 79.6·16-s + 105.·17-s + 35.2·19-s + 21.7·20-s + 92.0·22-s − 62.0·23-s − 102.·25-s + 157.·26-s − 145.·28-s + 249.·29-s − 129.·31-s − 186.·32-s + 375.·34-s − 147.·35-s − 219.·37-s + 125.·38-s − 56.4·40-s − 501.·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.577·4-s + 0.420·5-s − 1.69·7-s − 0.530·8-s + 0.528·10-s + 0.710·11-s + 0.944·13-s − 2.13·14-s − 1.24·16-s + 1.50·17-s + 0.425·19-s + 0.243·20-s + 0.891·22-s − 0.562·23-s − 0.822·25-s + 1.18·26-s − 0.981·28-s + 1.59·29-s − 0.748·31-s − 1.03·32-s + 1.89·34-s − 0.714·35-s − 0.976·37-s + 0.533·38-s − 0.223·40-s − 1.90·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 3.55T + 8T^{2} \) |
| 5 | \( 1 - 4.70T + 125T^{2} \) |
| 7 | \( 1 + 31.4T + 343T^{2} \) |
| 11 | \( 1 - 25.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 249.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 501.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 321.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 682.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 576.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 217.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 20.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 164.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 179.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 784.T + 9.12e5T^{2} \) |
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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
−8.449069258390703310267868294716, −7.17368801456369280932658374655, −6.39097029596386544717324600308, −5.96117898476641982547666955290, −5.27859197315671655281746110682, −4.03457264963509970158699429202, −3.46037108356065324523101617464, −2.88671392259014884192109106441, −1.40861422016824975237921477098, 0,
1.40861422016824975237921477098, 2.88671392259014884192109106441, 3.46037108356065324523101617464, 4.03457264963509970158699429202, 5.27859197315671655281746110682, 5.96117898476641982547666955290, 6.39097029596386544717324600308, 7.17368801456369280932658374655, 8.449069258390703310267868294716
