L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 4·11-s + 13-s − 15-s + 17-s + 6·19-s + 2·21-s + 8·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s − 2·35-s − 2·37-s − 39-s − 2·41-s − 10·43-s + 45-s − 4·47-s − 3·49-s − 51-s − 2·53-s + 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.37·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.338·35-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 1.52·43-s + 0.149·45-s − 0.583·47-s − 3/7·49-s − 0.140·51-s − 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.54519151717438, −15.11548265884716, −14.43040191303466, −13.90191100115585, −13.45037427612094, −12.82259281640883, −12.33069284783054, −11.85584371435537, −11.17987398135071, −10.84221111251879, −9.995546056687953, −9.589449544372483, −9.165969942120930, −8.570563439408966, −7.696507523516414, −6.970668462854148, −6.627574952671760, −6.131253461912776, −5.284806803811820, −4.994258145978015, −4.037919701467787, −3.287484705570440, −2.924689258885288, −1.517473588028201, −1.227549371598824, 0,
1.227549371598824, 1.517473588028201, 2.924689258885288, 3.287484705570440, 4.037919701467787, 4.994258145978015, 5.284806803811820, 6.131253461912776, 6.627574952671760, 6.970668462854148, 7.696507523516414, 8.570563439408966, 9.165969942120930, 9.589449544372483, 9.995546056687953, 10.84221111251879, 11.17987398135071, 11.85584371435537, 12.33069284783054, 12.82259281640883, 13.45037427612094, 13.90191100115585, 14.43040191303466, 15.11548265884716, 15.54519151717438