L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 7-s − 2·9-s + 2·10-s − 3·11-s + 2·12-s + 3·13-s + 2·14-s + 15-s − 4·16-s − 17-s − 4·18-s − 6·19-s + 2·20-s + 21-s − 6·22-s − 2·23-s + 25-s + 6·26-s − 5·27-s + 2·28-s − 3·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s − 2/3·9-s + 0.632·10-s − 0.904·11-s + 0.577·12-s + 0.832·13-s + 0.534·14-s + 0.258·15-s − 16-s − 0.242·17-s − 0.942·18-s − 1.37·19-s + 0.447·20-s + 0.218·21-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 1.17·26-s − 0.962·27-s + 0.377·28-s − 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.40609303321488918737352136002, −6.39729820283455888127193725754, −6.05547334061216319470775397163, −5.23644784185415968431036583531, −4.77851124132374451098118288248, −3.77656060833449355859079744944, −3.32873751151844003369048756236, −2.36360758653652029637163980202, −1.90024837111624115231112726687, 0,
1.90024837111624115231112726687, 2.36360758653652029637163980202, 3.32873751151844003369048756236, 3.77656060833449355859079744944, 4.77851124132374451098118288248, 5.23644784185415968431036583531, 6.05547334061216319470775397163, 6.39729820283455888127193725754, 7.40609303321488918737352136002