| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 2·7-s + 4·8-s + 3·9-s − 10·11-s + 6·12-s − 2·13-s + 4·14-s + 5·16-s − 10·17-s + 6·18-s − 10·19-s + 4·21-s − 20·22-s − 6·23-s + 8·24-s − 4·26-s + 4·27-s + 6·28-s + 4·31-s + 6·32-s − 20·33-s − 20·34-s + 9·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 3.01·11-s + 1.73·12-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 2.42·17-s + 1.41·18-s − 2.29·19-s + 0.872·21-s − 4.26·22-s − 1.25·23-s + 1.63·24-s − 0.784·26-s + 0.769·27-s + 1.13·28-s + 0.718·31-s + 1.06·32-s − 3.48·33-s − 3.42·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 204 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 77 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.943860279374787397609536859462, −7.75142775167152931174962864583, −7.24379331343296561611113617713, −6.86186758271696455865913868569, −6.41471508263481316992137059472, −6.34558239713159400990190468701, −5.65540445595124752635403095278, −5.31462586621855698817703645934, −4.84380782182716780208563681983, −4.64006518257924221585001556060, −4.32516918556379983574746213474, −4.16986578668810710869832876083, −3.29086307180216411739380712047, −3.09263472157620499758312176020, −2.44043439954830776966115344835, −2.41351218012830393414554719987, −1.84835139734570715143147531842, −1.79730687096142690278464728186, 0, 0,
1.79730687096142690278464728186, 1.84835139734570715143147531842, 2.41351218012830393414554719987, 2.44043439954830776966115344835, 3.09263472157620499758312176020, 3.29086307180216411739380712047, 4.16986578668810710869832876083, 4.32516918556379983574746213474, 4.64006518257924221585001556060, 4.84380782182716780208563681983, 5.31462586621855698817703645934, 5.65540445595124752635403095278, 6.34558239713159400990190468701, 6.41471508263481316992137059472, 6.86186758271696455865913868569, 7.24379331343296561611113617713, 7.75142775167152931174962864583, 7.943860279374787397609536859462
