| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s − 5-s + 6·6-s − 4·8-s + 3·9-s + 2·10-s − 12·11-s − 9·12-s + 3·15-s + 5·16-s + 4·17-s − 6·18-s + 4·19-s − 3·20-s + 24·22-s + 12·24-s − 9·29-s − 6·30-s − 4·31-s − 6·32-s + 36·33-s − 8·34-s + 9·36-s + 8·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s − 0.447·5-s + 2.44·6-s − 1.41·8-s + 9-s + 0.632·10-s − 3.61·11-s − 2.59·12-s + 0.774·15-s + 5/4·16-s + 0.970·17-s − 1.41·18-s + 0.917·19-s − 0.670·20-s + 5.11·22-s + 2.44·24-s − 1.67·29-s − 1.09·30-s − 0.718·31-s − 1.06·32-s + 6.26·33-s − 1.37·34-s + 3/2·36-s + 1.31·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 43 | $C_2$ | \( 1 + 13 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−10.82329198021321592103326677486, −10.68806091808231158796079668089, −10.11740796518777914487880407003, −9.647153288400384047556688445816, −9.472298557554296097755407749912, −8.292209352422757863040016787814, −8.249860879405008774862463864305, −7.70392866215363538328473552958, −7.34390571778061140995842898880, −7.03941804960068795595043853125, −5.94134191805903154512109648994, −5.78867838504659961623646506543, −5.25767330633036476134860173760, −5.14456401471591198803257239420, −4.03152599555149067529829374220, −2.83020870303270225348206797819, −2.77937321113669562370556187255, −1.44056956029966402223138470886, 0, 0,
1.44056956029966402223138470886, 2.77937321113669562370556187255, 2.83020870303270225348206797819, 4.03152599555149067529829374220, 5.14456401471591198803257239420, 5.25767330633036476134860173760, 5.78867838504659961623646506543, 5.94134191805903154512109648994, 7.03941804960068795595043853125, 7.34390571778061140995842898880, 7.70392866215363538328473552958, 8.249860879405008774862463864305, 8.292209352422757863040016787814, 9.472298557554296097755407749912, 9.647153288400384047556688445816, 10.11740796518777914487880407003, 10.68806091808231158796079668089, 10.82329198021321592103326677486
