| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 7·13-s − 15-s − 17-s − 7·19-s − 21-s − 2·23-s + 25-s + 27-s − 9·29-s + 35-s − 7·37-s − 7·39-s − 12·41-s − 12·43-s − 45-s − 2·47-s − 6·49-s − 51-s + 6·53-s − 7·57-s + 10·59-s − 2·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.94·13-s − 0.258·15-s − 0.242·17-s − 1.60·19-s − 0.218·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s + 0.169·35-s − 1.15·37-s − 1.12·39-s − 1.87·41-s − 1.82·43-s − 0.149·45-s − 0.291·47-s − 6/7·49-s − 0.140·51-s + 0.824·53-s − 0.927·57-s + 1.30·59-s − 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2720873582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2720873582\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.04914152792760, −14.84604402422854, −14.27179985270781, −13.41288823533479, −13.18712546034142, −12.43117148610210, −12.15413588184009, −11.50442896871394, −10.84229356822432, −10.10656815936337, −9.879458023826683, −9.212819552532196, −8.507107563553241, −8.173054748859581, −7.457761941279461, −6.788160326977065, −6.652872479957453, −5.447839138752963, −5.010283626682402, −4.283167883083575, −3.664466988310711, −3.061462616708811, −2.136319108936724, −1.845264278782333, −0.1774081050469374,
0.1774081050469374, 1.845264278782333, 2.136319108936724, 3.061462616708811, 3.664466988310711, 4.283167883083575, 5.010283626682402, 5.447839138752963, 6.652872479957453, 6.788160326977065, 7.457761941279461, 8.173054748859581, 8.507107563553241, 9.212819552532196, 9.879458023826683, 10.10656815936337, 10.84229356822432, 11.50442896871394, 12.15413588184009, 12.43117148610210, 13.18712546034142, 13.41288823533479, 14.27179985270781, 14.84604402422854, 15.04914152792760
