| L(s) = 1 | − 2·2-s − 4-s − 4·5-s + 8·8-s + 8·10-s − 4·11-s + 6·13-s − 7·16-s − 2·17-s − 4·19-s + 4·20-s + 8·22-s − 12·23-s + 11·25-s − 12·26-s − 8·31-s − 14·32-s + 4·34-s + 8·38-s − 32·40-s − 10·41-s − 8·43-s + 4·44-s + 24·46-s − 2·49-s − 22·50-s − 6·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 1.78·5-s + 2.82·8-s + 2.52·10-s − 1.20·11-s + 1.66·13-s − 7/4·16-s − 0.485·17-s − 0.917·19-s + 0.894·20-s + 1.70·22-s − 2.50·23-s + 11/5·25-s − 2.35·26-s − 1.43·31-s − 2.47·32-s + 0.685·34-s + 1.29·38-s − 5.05·40-s − 1.56·41-s − 1.21·43-s + 0.603·44-s + 3.53·46-s − 2/7·49-s − 3.11·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.44016326952000976763663221571, −10.19580472138636138727025828670, −9.322859735762386429750280719948, −9.234442894354025151005899457583, −8.431823031171535051611908253890, −8.257647050179503332413616272046, −8.117163521023684553224208364334, −7.86475128922905679153798779370, −6.96616124658637756845645015965, −6.87149501768392047479566463655, −5.84532834263025224948276154556, −5.35750253168669156142434462081, −4.68796683712046118144976040226, −4.25041080791483631842875949300, −3.70723719805815684549717391530, −3.57286889013146522997574076602, −2.17813737496736677871601431537, −1.35755502174332698773915051828, 0, 0,
1.35755502174332698773915051828, 2.17813737496736677871601431537, 3.57286889013146522997574076602, 3.70723719805815684549717391530, 4.25041080791483631842875949300, 4.68796683712046118144976040226, 5.35750253168669156142434462081, 5.84532834263025224948276154556, 6.87149501768392047479566463655, 6.96616124658637756845645015965, 7.86475128922905679153798779370, 8.117163521023684553224208364334, 8.257647050179503332413616272046, 8.431823031171535051611908253890, 9.234442894354025151005899457583, 9.322859735762386429750280719948, 10.19580472138636138727025828670, 10.44016326952000976763663221571
