| L(s) = 1 | − 3-s − 3·5-s − 2·9-s − 4·11-s + 8·13-s + 3·15-s − 8·19-s − 4·23-s + 2·25-s + 2·27-s − 16·29-s + 4·33-s − 8·39-s + 4·41-s − 4·43-s + 6·45-s + 12·47-s − 14·49-s + 8·53-s + 12·55-s + 8·57-s − 12·59-s + 4·61-s − 24·65-s + 12·67-s + 4·69-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 2/3·9-s − 1.20·11-s + 2.21·13-s + 0.774·15-s − 1.83·19-s − 0.834·23-s + 2/5·25-s + 0.384·27-s − 2.97·29-s + 0.696·33-s − 1.28·39-s + 0.624·41-s − 0.609·43-s + 0.894·45-s + 1.75·47-s − 2·49-s + 1.09·53-s + 1.61·55-s + 1.05·57-s − 1.56·59-s + 0.512·61-s − 2.97·65-s + 1.46·67-s + 0.481·69-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−16.2652641967, −15.7488207479, −15.5114390469, −14.9757145558, −14.5765256398, −13.7392025398, −13.2768755254, −12.9418318979, −12.3574631320, −11.7666126827, −11.1766293089, −10.9076921437, −10.7974752774, −9.83549734579, −8.95538623117, −8.58268357776, −7.88889261724, −7.77199473906, −6.76366111245, −5.87146418849, −5.86487938441, −4.78210199118, −3.76484405046, −3.67478222653, −2.15654793578, 0,
2.15654793578, 3.67478222653, 3.76484405046, 4.78210199118, 5.86487938441, 5.87146418849, 6.76366111245, 7.77199473906, 7.88889261724, 8.58268357776, 8.95538623117, 9.83549734579, 10.7974752774, 10.9076921437, 11.1766293089, 11.7666126827, 12.3574631320, 12.9418318979, 13.2768755254, 13.7392025398, 14.5765256398, 14.9757145558, 15.5114390469, 15.7488207479, 16.2652641967
