| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 6·11-s + 13-s + 14-s + 16-s + 3·17-s − 4·19-s + 6·22-s + 3·23-s − 26-s − 28-s − 3·29-s + 5·31-s − 32-s − 3·34-s + 4·37-s + 4·38-s − 6·41-s + 7·43-s − 6·44-s − 3·46-s − 6·47-s + 49-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s + 1.27·22-s + 0.625·23-s − 0.196·26-s − 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 0.657·37-s + 0.648·38-s − 0.937·41-s + 1.06·43-s − 0.904·44-s − 0.442·46-s − 0.875·47-s + 1/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8296866541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8296866541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.65188927126179618202621928663, −7.29958357420562174652635461537, −6.33523644926906785755281943189, −5.78343260342706907108390979814, −5.05438564133687495354976376831, −4.20875290441993744328792800999, −3.09182089458528085823268828858, −2.67728807910800557655070325428, −1.65257932458882019625186593173, −0.47599076617655209228444775321,
0.47599076617655209228444775321, 1.65257932458882019625186593173, 2.67728807910800557655070325428, 3.09182089458528085823268828858, 4.20875290441993744328792800999, 5.05438564133687495354976376831, 5.78343260342706907108390979814, 6.33523644926906785755281943189, 7.29958357420562174652635461537, 7.65188927126179618202621928663
