L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 13-s + 2·14-s − 15-s + 16-s − 18-s − 2·19-s + 20-s + 2·21-s + 4·22-s − 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·28-s + 8·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s + 0.852·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.66556287769163, −12.18926674882286, −11.88105631812216, −11.22499103551848, −10.70415369045860, −10.48634939084488, −9.978940570994774, −9.647063683750186, −9.315599391333241, −8.400051754019732, −8.289880536727088, −7.821469010819056, −7.012311676922416, −6.725430209458124, −6.435323321668111, −5.738121238478699, −5.386603454818694, −4.845280454886660, −4.364544688301269, −3.487846869017549, −3.099976062331866, −2.444995529380191, −2.033118559452689, −1.309528657244285, −0.5421319677240333, 0,
0.5421319677240333, 1.309528657244285, 2.033118559452689, 2.444995529380191, 3.099976062331866, 3.487846869017549, 4.364544688301269, 4.845280454886660, 5.386603454818694, 5.738121238478699, 6.435323321668111, 6.725430209458124, 7.012311676922416, 7.821469010819056, 8.289880536727088, 8.400051754019732, 9.315599391333241, 9.647063683750186, 9.978940570994774, 10.48634939084488, 10.70415369045860, 11.22499103551848, 11.88105631812216, 12.18926674882286, 12.66556287769163