| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 6·9-s + 4·10-s + 4·13-s − 4·16-s + 16·17-s − 12·18-s + 4·20-s − 25-s + 8·26-s − 8·29-s − 8·32-s + 32·34-s − 12·36-s + 10·37-s + 18·41-s − 12·45-s − 2·50-s + 8·52-s − 16·58-s − 20·61-s − 8·64-s + 8·65-s + 32·68-s − 22·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 2·9-s + 1.26·10-s + 1.10·13-s − 16-s + 3.88·17-s − 2.82·18-s + 0.894·20-s − 1/5·25-s + 1.56·26-s − 1.48·29-s − 1.41·32-s + 5.48·34-s − 2·36-s + 1.64·37-s + 2.81·41-s − 1.78·45-s − 0.282·50-s + 1.10·52-s − 2.10·58-s − 2.56·61-s − 64-s + 0.992·65-s + 3.88·68-s − 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.238497722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.238497722\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−12.39172627197990640816249023358, −11.84029226237260504818714605529, −11.40821634262863927966098851811, −11.03696599470940776007965394579, −10.50990469803050651780308257422, −9.799173258447029003936865600607, −9.248664173205954451222947799788, −9.151926648514855079848796927288, −8.044448484150280631084870186537, −7.971289347862400977582418576470, −7.28341722062754824078469859375, −6.17398438869696052795988925966, −5.85118017469234000821358836988, −5.61869487076155767706723202651, −5.52640730083242909696074743816, −4.35912269837072031073867218347, −3.67841135151981855054211735185, −2.99732352108390502275135572177, −2.76082817555967707330451924694, −1.35404777805397397411721894086,
1.35404777805397397411721894086, 2.76082817555967707330451924694, 2.99732352108390502275135572177, 3.67841135151981855054211735185, 4.35912269837072031073867218347, 5.52640730083242909696074743816, 5.61869487076155767706723202651, 5.85118017469234000821358836988, 6.17398438869696052795988925966, 7.28341722062754824078469859375, 7.971289347862400977582418576470, 8.044448484150280631084870186537, 9.151926648514855079848796927288, 9.248664173205954451222947799788, 9.799173258447029003936865600607, 10.50990469803050651780308257422, 11.03696599470940776007965394579, 11.40821634262863927966098851811, 11.84029226237260504818714605529, 12.39172627197990640816249023358
