| L(s) = 1 | + 2-s + 4-s − 4.11·5-s − 3.78·7-s + 8-s − 4.11·10-s − 2.15·11-s + 5.55·13-s − 3.78·14-s + 16-s + 2.24·17-s + 2.44·19-s − 4.11·20-s − 2.15·22-s + 0.398·23-s + 11.8·25-s + 5.55·26-s − 3.78·28-s − 8.12·29-s + 0.810·31-s + 32-s + 2.24·34-s + 15.5·35-s + 8.17·37-s + 2.44·38-s − 4.11·40-s − 0.726·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.83·5-s − 1.43·7-s + 0.353·8-s − 1.29·10-s − 0.649·11-s + 1.54·13-s − 1.01·14-s + 0.250·16-s + 0.545·17-s + 0.560·19-s − 0.919·20-s − 0.458·22-s + 0.0831·23-s + 2.37·25-s + 1.08·26-s − 0.715·28-s − 1.50·29-s + 0.145·31-s + 0.176·32-s + 0.385·34-s + 2.62·35-s + 1.34·37-s + 0.396·38-s − 0.649·40-s − 0.113·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 4.11T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 - 0.398T + 23T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 - 0.810T + 31T^{2} \) |
| 37 | \( 1 - 8.17T + 37T^{2} \) |
| 41 | \( 1 + 0.726T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 - 9.53T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.54T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 8.38T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 1.42T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.25603718617755490337247684353, −6.95991865479497761249870380588, −6.00052006147924028227822493437, −5.47555885781706027378821121651, −4.41046332455332221864995928305, −3.72007947328614174091278935790, −3.41693971215935144349859288992, −2.70760300503221383509143544314, −1.08863223972978654976562379060, 0,
1.08863223972978654976562379060, 2.70760300503221383509143544314, 3.41693971215935144349859288992, 3.72007947328614174091278935790, 4.41046332455332221864995928305, 5.47555885781706027378821121651, 6.00052006147924028227822493437, 6.95991865479497761249870380588, 7.25603718617755490337247684353
