| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 4·11-s − 12-s − 13-s + 15-s + 16-s + 3·17-s + 18-s − 4·19-s − 20-s − 4·22-s − 24-s − 4·25-s − 26-s − 27-s − 6·29-s + 30-s − 6·31-s + 32-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.07·31-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.40302229855070, −12.99640029995197, −12.51573856428122, −12.35858210669861, −11.58204790491843, −11.23138844231829, −10.87001291414852, −10.28739461336224, −9.934413598406095, −9.293613104662486, −8.669789925970515, −7.906400825374529, −7.727808421174062, −7.196009161117676, −6.673782251972989, −5.866463085213187, −5.701377300593837, −5.138010364618216, −4.604337677721589, −4.016626840594643, −3.561819480417853, −2.922160555648755, −2.165432317070131, −1.773884624032203, −0.6816781694852415, 0,
0.6816781694852415, 1.773884624032203, 2.165432317070131, 2.922160555648755, 3.561819480417853, 4.016626840594643, 4.604337677721589, 5.138010364618216, 5.701377300593837, 5.866463085213187, 6.673782251972989, 7.196009161117676, 7.727808421174062, 7.906400825374529, 8.669789925970515, 9.293613104662486, 9.934413598406095, 10.28739461336224, 10.87001291414852, 11.23138844231829, 11.58204790491843, 12.35858210669861, 12.51573856428122, 12.99640029995197, 13.40302229855070
