| L(s) = 1 | + 2-s − 3·3-s − 3·6-s − 7-s − 8-s + 6·9-s − 6·11-s + 2·13-s − 14-s − 16-s + 6·18-s − 8·19-s + 3·21-s − 6·22-s + 9·23-s + 3·24-s + 2·26-s − 9·27-s − 3·29-s + 4·31-s + 18·33-s − 16·37-s − 8·38-s − 6·39-s + 3·41-s + 3·42-s + 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 1.80·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.41·18-s − 1.83·19-s + 0.654·21-s − 1.27·22-s + 1.87·23-s + 0.612·24-s + 0.392·26-s − 1.73·27-s − 0.557·29-s + 0.718·31-s + 3.13·33-s − 2.63·37-s − 1.29·38-s − 0.960·39-s + 0.468·41-s + 0.462·42-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6448048862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6448048862\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.17925332206604171150886318411, −10.84099521265143671295328440686, −10.57854249578760389817887648176, −10.38884765973097015677744511743, −9.704810257872044344521578935686, −9.018186212773898703791761130056, −8.703790023742109870838687063123, −8.043822834862214407046526016415, −7.42293829254978701231434195071, −6.98324815420559723364351552938, −6.44361951147323624871802717048, −6.11906377353399424568055431716, −5.43896921001697145723242181190, −5.26922934678786824812404839113, −4.65604491669737574417185590289, −4.28338363068696487602464209285, −3.43046685298556412701752143066, −2.79002133094681193256273376833, −1.83284332040228950710394648830, −0.49136591243944856185936137948,
0.49136591243944856185936137948, 1.83284332040228950710394648830, 2.79002133094681193256273376833, 3.43046685298556412701752143066, 4.28338363068696487602464209285, 4.65604491669737574417185590289, 5.26922934678786824812404839113, 5.43896921001697145723242181190, 6.11906377353399424568055431716, 6.44361951147323624871802717048, 6.98324815420559723364351552938, 7.42293829254978701231434195071, 8.043822834862214407046526016415, 8.703790023742109870838687063123, 9.018186212773898703791761130056, 9.704810257872044344521578935686, 10.38884765973097015677744511743, 10.57854249578760389817887648176, 10.84099521265143671295328440686, 11.17925332206604171150886318411
