L(s) = 1 | − 3-s − 5-s + 9-s + 11-s − 13-s + 15-s − 2·19-s + 9·23-s + 25-s − 27-s − 9·29-s − 8·31-s − 33-s + 2·37-s + 39-s + 3·41-s + 43-s − 45-s − 9·47-s + 9·53-s − 55-s + 2·57-s − 12·59-s − 10·61-s + 65-s + 4·67-s − 9·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.458·19-s + 1.87·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 1.43·31-s − 0.174·33-s + 0.328·37-s + 0.160·39-s + 0.468·41-s + 0.152·43-s − 0.149·45-s − 1.31·47-s + 1.23·53-s − 0.134·55-s + 0.264·57-s − 1.56·59-s − 1.28·61-s + 0.124·65-s + 0.488·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.58228594358913, −13.10458683347545, −12.75180396397433, −12.33890355159114, −11.77337400924283, −11.13866157997573, −11.00918181818343, −10.61081198870670, −9.727663355378745, −9.392620626548646, −8.945710513239769, −8.373872837359636, −7.680753280615828, −7.230105607209640, −6.942425478130578, −6.248580540651332, −5.646874366128861, −5.280380975450115, −4.537667838438320, −4.243132604608045, −3.394123280608540, −3.085312170732294, −2.091134374989174, −1.562425734679257, −0.7251315502423034, 0,
0.7251315502423034, 1.562425734679257, 2.091134374989174, 3.085312170732294, 3.394123280608540, 4.243132604608045, 4.537667838438320, 5.280380975450115, 5.646874366128861, 6.248580540651332, 6.942425478130578, 7.230105607209640, 7.680753280615828, 8.373872837359636, 8.945710513239769, 9.392620626548646, 9.727663355378745, 10.61081198870670, 11.00918181818343, 11.13866157997573, 11.77337400924283, 12.33890355159114, 12.75180396397433, 13.10458683347545, 13.58228594358913