| L(s) = 1 | − 2·3-s − 2·7-s − 3·9-s + 2·11-s + 8·13-s + 4·17-s − 2·19-s + 4·21-s − 2·23-s + 14·27-s − 10·29-s − 4·31-s − 4·33-s + 6·37-s − 16·39-s − 6·41-s − 6·43-s + 10·47-s − 6·49-s − 8·51-s + 12·53-s + 4·57-s + 2·61-s + 6·63-s + 6·67-s + 4·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s − 9-s + 0.603·11-s + 2.21·13-s + 0.970·17-s − 0.458·19-s + 0.872·21-s − 0.417·23-s + 2.69·27-s − 1.85·29-s − 0.718·31-s − 0.696·33-s + 0.986·37-s − 2.56·39-s − 0.937·41-s − 0.914·43-s + 1.45·47-s − 6/7·49-s − 1.12·51-s + 1.64·53-s + 0.529·57-s + 0.256·61-s + 0.755·63-s + 0.733·67-s + 0.481·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061989523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061989523\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 274 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.78612874515693292427092827127, −7.67342451891118470709211023704, −6.88961700328663164166566288243, −6.84311409760534895451343174887, −6.32438884561861335552826368026, −6.23286776968904553697651828431, −5.75628705987364156242274840039, −5.67169514826287205137954092813, −5.21016990197140574644002212676, −5.14698742909940593088810783346, −4.10804979127148871808135967824, −4.10274851193734865322183241497, −3.63363231020048242019119812360, −3.43860959492098685767845036764, −2.76777188983624469780089030927, −2.63388176087843319606250870136, −1.68523737327056418837228435891, −1.49815213726767688694415220770, −0.77865746089978186362120883757, −0.34501124962802038699916582977,
0.34501124962802038699916582977, 0.77865746089978186362120883757, 1.49815213726767688694415220770, 1.68523737327056418837228435891, 2.63388176087843319606250870136, 2.76777188983624469780089030927, 3.43860959492098685767845036764, 3.63363231020048242019119812360, 4.10274851193734865322183241497, 4.10804979127148871808135967824, 5.14698742909940593088810783346, 5.21016990197140574644002212676, 5.67169514826287205137954092813, 5.75628705987364156242274840039, 6.23286776968904553697651828431, 6.32438884561861335552826368026, 6.84311409760534895451343174887, 6.88961700328663164166566288243, 7.67342451891118470709211023704, 7.78612874515693292427092827127
