| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s + 2·13-s + 2·17-s + 8·23-s + 3·25-s − 4·29-s − 4·35-s + 12·37-s − 12·41-s + 2·43-s − 8·47-s + 6·49-s − 4·55-s − 8·59-s + 20·61-s + 4·65-s + 4·67-s − 4·71-s + 18·73-s + 4·77-s − 12·79-s + 2·83-s + 4·85-s + 16·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 1.66·23-s + 3/5·25-s − 0.742·29-s − 0.676·35-s + 1.97·37-s − 1.87·41-s + 0.304·43-s − 1.16·47-s + 6/7·49-s − 0.539·55-s − 1.04·59-s + 2.56·61-s + 0.496·65-s + 0.488·67-s − 0.474·71-s + 2.10·73-s + 0.455·77-s − 1.35·79-s + 0.219·83-s + 0.433·85-s + 1.69·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.090471188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.090471188\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.681927373356142504717409803471, −8.237266220587868450622337279627, −8.008151874342029604760356293887, −7.52444327848299020739194379237, −6.99534207064231104535886260551, −6.90183151129377935470332399835, −6.33194746375446376072284713970, −6.16702117471608079278945832342, −5.59790009145838515672466575544, −5.41082390874012797406560280836, −4.86482019649560809189618081266, −4.73108754265234899709649356983, −3.81727508144634279978041275167, −3.71360024776516049312484128673, −2.99048750587129398976343581364, −2.89622172001196878056084332535, −2.22995185898318267104759345975, −1.78009553567555265826743142776, −1.07440916665369499545697458968, −0.56887589256857787452940651825,
0.56887589256857787452940651825, 1.07440916665369499545697458968, 1.78009553567555265826743142776, 2.22995185898318267104759345975, 2.89622172001196878056084332535, 2.99048750587129398976343581364, 3.71360024776516049312484128673, 3.81727508144634279978041275167, 4.73108754265234899709649356983, 4.86482019649560809189618081266, 5.41082390874012797406560280836, 5.59790009145838515672466575544, 6.16702117471608079278945832342, 6.33194746375446376072284713970, 6.90183151129377935470332399835, 6.99534207064231104535886260551, 7.52444327848299020739194379237, 8.008151874342029604760356293887, 8.237266220587868450622337279627, 8.681927373356142504717409803471
