| L(s) = 1 | + 2·3-s + 2·5-s + 3·9-s + 4·15-s − 8·19-s + 3·25-s + 4·27-s − 4·31-s + 6·45-s + 14·49-s − 16·57-s − 12·59-s + 20·61-s + 8·67-s − 28·73-s + 6·75-s + 28·79-s + 5·81-s − 8·93-s − 16·95-s − 12·101-s − 8·103-s − 24·107-s + 10·121-s + 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 9-s + 1.03·15-s − 1.83·19-s + 3/5·25-s + 0.769·27-s − 0.718·31-s + 0.894·45-s + 2·49-s − 2.11·57-s − 1.56·59-s + 2.56·61-s + 0.977·67-s − 3.27·73-s + 0.692·75-s + 3.15·79-s + 5/9·81-s − 0.829·93-s − 1.64·95-s − 1.19·101-s − 0.788·103-s − 2.32·107-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.681223997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.681223997\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.476774708166337686592410230821, −8.245036317676522673683277458715, −7.87787115486009899093262521863, −7.51388952513866013267515263990, −6.95926625021691650094962106942, −6.72575109735797958936065707001, −6.52634394994874698322262885455, −5.93047201228314773852540650754, −5.51576277471748180185955958968, −5.34552681182333840847444051958, −4.68546231262043105634160233819, −4.23620559613164206393440062539, −4.03525711104170151474183718575, −3.58311703895617436239772349244, −2.97477243240723019796176203917, −2.61610068872793840162000454166, −2.18985871907087045381540913148, −1.87193557791918760841205381417, −1.33240923041998345813823778269, −0.53159866053781738224170611867,
0.53159866053781738224170611867, 1.33240923041998345813823778269, 1.87193557791918760841205381417, 2.18985871907087045381540913148, 2.61610068872793840162000454166, 2.97477243240723019796176203917, 3.58311703895617436239772349244, 4.03525711104170151474183718575, 4.23620559613164206393440062539, 4.68546231262043105634160233819, 5.34552681182333840847444051958, 5.51576277471748180185955958968, 5.93047201228314773852540650754, 6.52634394994874698322262885455, 6.72575109735797958936065707001, 6.95926625021691650094962106942, 7.51388952513866013267515263990, 7.87787115486009899093262521863, 8.245036317676522673683277458715, 8.476774708166337686592410230821
