| L(s) = 1 | + 0.246·2-s − 1.93·4-s + 1.69·7-s − 0.972·8-s + 0.911·11-s + 1.55·13-s + 0.417·14-s + 3.63·16-s − 5.29·17-s − 19-s + 0.225·22-s − 4.24·23-s + 0.384·26-s − 3.28·28-s − 5.00·29-s + 1.82·31-s + 2.84·32-s − 1.30·34-s + 6.29·37-s − 0.246·38-s − 4.18·41-s + 7.31·43-s − 1.76·44-s − 1.04·46-s + 2.04·47-s − 4.13·49-s − 3.01·52-s + ⋯ |
| L(s) = 1 | + 0.174·2-s − 0.969·4-s + 0.639·7-s − 0.343·8-s + 0.274·11-s + 0.431·13-s + 0.111·14-s + 0.909·16-s − 1.28·17-s − 0.229·19-s + 0.0480·22-s − 0.885·23-s + 0.0753·26-s − 0.620·28-s − 0.930·29-s + 0.328·31-s + 0.502·32-s − 0.224·34-s + 1.03·37-s − 0.0400·38-s − 0.652·41-s + 1.11·43-s − 0.266·44-s − 0.154·46-s + 0.298·47-s − 0.591·49-s − 0.418·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.246T + 2T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 0.911T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 0.542T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 - 1.59T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 - 1.55T + 97T^{2} \) |
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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
−8.107194304989024981418223306551, −7.46082487381810938075252772941, −6.36085002128734901545080700788, −5.82015000606447604151477099607, −4.85016213006715481697956828238, −4.30285423089235161442316339831, −3.64630473049700540338608358491, −2.43657219240356756124696537364, −1.35707431756539852706439668830, 0,
1.35707431756539852706439668830, 2.43657219240356756124696537364, 3.64630473049700540338608358491, 4.30285423089235161442316339831, 4.85016213006715481697956828238, 5.82015000606447604151477099607, 6.36085002128734901545080700788, 7.46082487381810938075252772941, 8.107194304989024981418223306551
