| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s − 4·11-s − 6·12-s + 5·16-s + 8·17-s + 6·18-s + 8·19-s − 8·22-s − 12·23-s − 8·24-s − 4·27-s + 8·29-s + 12·31-s + 6·32-s + 8·33-s + 16·34-s + 9·36-s − 8·37-s + 16·38-s − 4·41-s − 4·43-s − 12·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 9-s − 1.20·11-s − 1.73·12-s + 5/4·16-s + 1.94·17-s + 1.41·18-s + 1.83·19-s − 1.70·22-s − 2.50·23-s − 1.63·24-s − 0.769·27-s + 1.48·29-s + 2.15·31-s + 1.06·32-s + 1.39·33-s + 2.74·34-s + 3/2·36-s − 1.31·37-s + 2.59·38-s − 0.624·41-s − 0.609·43-s − 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.070485705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.070485705\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 88 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.943847069632229757651457100246, −7.74891775850857897164559316990, −7.27937661176114399878793193550, −6.79151434475101125172843549267, −6.54715151123825976392799486638, −6.33091701021095749697933958644, −5.72481476920333027812440041738, −5.58249762861459552166960893998, −5.19542722202946478051492020695, −5.16544639333246706434988218372, −4.58041051025677143173928819918, −4.34192056010599681163142954855, −3.59132189096908773714632640797, −3.59119478652307199329994748015, −3.02224859191587521000171333634, −2.67680219011026840865841170826, −2.09152759847903227016656292491, −1.62132269306435964733202711054, −0.967378328829978976772560362377, −0.58336123294360954269633659691,
0.58336123294360954269633659691, 0.967378328829978976772560362377, 1.62132269306435964733202711054, 2.09152759847903227016656292491, 2.67680219011026840865841170826, 3.02224859191587521000171333634, 3.59119478652307199329994748015, 3.59132189096908773714632640797, 4.34192056010599681163142954855, 4.58041051025677143173928819918, 5.16544639333246706434988218372, 5.19542722202946478051492020695, 5.58249762861459552166960893998, 5.72481476920333027812440041738, 6.33091701021095749697933958644, 6.54715151123825976392799486638, 6.79151434475101125172843549267, 7.27937661176114399878793193550, 7.74891775850857897164559316990, 7.943847069632229757651457100246
