| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 3·9-s − 4·11-s + 5·16-s + 6·18-s − 8·22-s + 6·23-s + 25-s + 18·29-s + 6·32-s + 9·36-s − 8·37-s − 10·43-s − 12·44-s + 12·46-s + 2·50-s − 4·53-s + 36·58-s + 7·64-s − 18·67-s + 4·71-s + 12·72-s − 16·74-s + 20·79-s − 20·86-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 9-s − 1.20·11-s + 5/4·16-s + 1.41·18-s − 1.70·22-s + 1.25·23-s + 1/5·25-s + 3.34·29-s + 1.06·32-s + 3/2·36-s − 1.31·37-s − 1.52·43-s − 1.80·44-s + 1.76·46-s + 0.282·50-s − 0.549·53-s + 4.72·58-s + 7/8·64-s − 2.19·67-s + 0.474·71-s + 1.41·72-s − 1.85·74-s + 2.25·79-s − 2.15·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.628048522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.628048522\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.801329180936076378169644135886, −8.417995701997192590477084041039, −7.916325009885536263558334349206, −7.36597660738767186076247894677, −6.89394037901040901071472214534, −6.57205733073059424709937866086, −6.09157415694094013326272320913, −5.26789391708575690709750011159, −4.88294429143581850116350412354, −4.72473630829615747045956251199, −4.00544203434240366935428278549, −3.08854077694740486482752774267, −3.00212844655097191031458367831, −2.07618093352752431919693916301, −1.16479363083596500346798967246,
1.16479363083596500346798967246, 2.07618093352752431919693916301, 3.00212844655097191031458367831, 3.08854077694740486482752774267, 4.00544203434240366935428278549, 4.72473630829615747045956251199, 4.88294429143581850116350412354, 5.26789391708575690709750011159, 6.09157415694094013326272320913, 6.57205733073059424709937866086, 6.89394037901040901071472214534, 7.36597660738767186076247894677, 7.916325009885536263558334349206, 8.417995701997192590477084041039, 8.801329180936076378169644135886
