| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 4·7-s + 2·9-s + 4·10-s + 2·11-s − 2·13-s − 8·14-s − 4·16-s + 2·17-s + 4·18-s + 4·20-s + 4·22-s − 2·25-s − 4·26-s − 8·28-s − 6·29-s − 4·31-s − 8·32-s + 4·34-s − 8·35-s + 4·36-s + 2·37-s + 6·41-s − 6·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 1.51·7-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 0.554·13-s − 2.13·14-s − 16-s + 0.485·17-s + 0.942·18-s + 0.894·20-s + 0.852·22-s − 2/5·25-s − 0.784·26-s − 1.51·28-s − 1.11·29-s − 0.718·31-s − 1.41·32-s + 0.685·34-s − 1.35·35-s + 2/3·36-s + 0.328·37-s + 0.937·41-s − 0.914·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040570943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040570943\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−16.2129727883, −15.7596705952, −15.1836034845, −14.7523641276, −14.2738858087, −13.8006407370, −13.1988950177, −13.0497708481, −12.5745045680, −12.0535363849, −11.5920060950, −10.7809967502, −10.1469093856, −9.59268094738, −9.38081145469, −8.70867666988, −7.43257289770, −7.11899901691, −6.35557435124, −5.86715320870, −5.46707592359, −4.43199406230, −3.80805208567, −3.08918742679, −2.09261630544,
2.09261630544, 3.08918742679, 3.80805208567, 4.43199406230, 5.46707592359, 5.86715320870, 6.35557435124, 7.11899901691, 7.43257289770, 8.70867666988, 9.38081145469, 9.59268094738, 10.1469093856, 10.7809967502, 11.5920060950, 12.0535363849, 12.5745045680, 13.0497708481, 13.1988950177, 13.8006407370, 14.2738858087, 14.7523641276, 15.1836034845, 15.7596705952, 16.2129727883
