| L(s) = 1 | − 3-s − 5-s − 5·7-s + 9-s + 11-s − 13-s + 15-s + 17-s − 5·19-s + 5·21-s + 6·23-s + 25-s − 27-s − 3·29-s − 6·31-s − 33-s + 5·35-s + 37-s + 39-s + 7·41-s − 10·43-s − 45-s + 47-s + 18·49-s − 51-s + 9·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s − 1.14·19-s + 1.09·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 1.07·31-s − 0.174·33-s + 0.845·35-s + 0.164·37-s + 0.160·39-s + 1.09·41-s − 1.52·43-s − 0.149·45-s + 0.145·47-s + 18/7·49-s − 0.140·51-s + 1.23·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.51667628104337, −16.14407811523959, −15.38961887746292, −14.98039724349600, −14.41943205312161, −13.34047087677685, −13.06559417367977, −12.62264557448856, −12.07016546867969, −11.39414516976648, −10.79113512172746, −10.21343768358631, −9.639759627364658, −9.074005081137469, −8.519683555749080, −7.486715172562879, −6.937347406677735, −6.573958017184116, −5.823977159015435, −5.249301568927342, −4.252552536850809, −3.708121525766687, −3.042346618504199, −2.160076021601625, −0.8272681625737687, 0,
0.8272681625737687, 2.160076021601625, 3.042346618504199, 3.708121525766687, 4.252552536850809, 5.249301568927342, 5.823977159015435, 6.573958017184116, 6.937347406677735, 7.486715172562879, 8.519683555749080, 9.074005081137469, 9.639759627364658, 10.21343768358631, 10.79113512172746, 11.39414516976648, 12.07016546867969, 12.62264557448856, 13.06559417367977, 13.34047087677685, 14.41943205312161, 14.98039724349600, 15.38961887746292, 16.14407811523959, 16.51667628104337
