L(s) = 1 | − 2-s − 4-s + 4·5-s + 2·7-s + 3·8-s − 4·10-s + 11-s − 2·13-s − 2·14-s − 16-s − 2·17-s + 6·19-s − 4·20-s − 22-s + 11·25-s + 2·26-s − 2·28-s − 6·29-s + 4·31-s − 5·32-s + 2·34-s + 8·35-s + 6·37-s − 6·38-s + 12·40-s − 10·41-s − 6·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s + 0.755·7-s + 1.06·8-s − 1.26·10-s + 0.301·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.894·20-s − 0.213·22-s + 11/5·25-s + 0.392·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 1.35·35-s + 0.986·37-s − 0.973·38-s + 1.89·40-s − 1.56·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.58896177992500, −14.24249620300196, −13.54238052080117, −13.38129582194344, −12.97765473326721, −12.07711688289325, −11.57924368052024, −10.97340958331146, −10.42010217012633, −9.778569599722382, −9.639150460791499, −9.234951916144144, −8.439192003706032, −8.182280024875207, −7.366812183911304, −6.861886855934621, −6.240673985105784, −5.460738621083916, −5.113087515573725, −4.701851235233339, −3.829620838541927, −2.948522116723307, −2.217465608749423, −1.500214486549941, −1.233301204848757, 0,
1.233301204848757, 1.500214486549941, 2.217465608749423, 2.948522116723307, 3.829620838541927, 4.701851235233339, 5.113087515573725, 5.460738621083916, 6.240673985105784, 6.861886855934621, 7.366812183911304, 8.182280024875207, 8.439192003706032, 9.234951916144144, 9.639150460791499, 9.778569599722382, 10.42010217012633, 10.97340958331146, 11.57924368052024, 12.07711688289325, 12.97765473326721, 13.38129582194344, 13.54238052080117, 14.24249620300196, 14.58896177992500