| L(s) = 1 | + 2·2-s + 4·3-s + 3·4-s + 2·5-s + 8·6-s + 2·7-s + 4·8-s + 6·9-s + 4·10-s − 2·11-s + 12·12-s + 2·13-s + 4·14-s + 8·15-s + 5·16-s − 2·17-s + 12·18-s − 2·19-s + 6·20-s + 8·21-s − 4·22-s − 2·23-s + 16·24-s + 3·25-s + 4·26-s − 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 3/2·4-s + 0.894·5-s + 3.26·6-s + 0.755·7-s + 1.41·8-s + 2·9-s + 1.26·10-s − 0.603·11-s + 3.46·12-s + 0.554·13-s + 1.06·14-s + 2.06·15-s + 5/4·16-s − 0.485·17-s + 2.82·18-s − 0.458·19-s + 1.34·20-s + 1.74·21-s − 0.852·22-s − 0.417·23-s + 3.26·24-s + 3/5·25-s + 0.784·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.90188090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.90188090\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−10.55348405566603840281525483099, −10.14783602097039943396055759284, −9.507044163423851554345460825530, −9.259740009163592005687544437399, −8.589571810335369186252568363816, −8.422076705731661692957613300515, −8.094738857572422564011297863893, −7.57892340876491415450271085909, −7.11335892828630119795790234068, −6.50910631703067432125052856275, −6.11423148148720500758859467564, −5.52216342191513768986411681788, −5.06059656390981273772252475504, −4.61521477311126278475137722645, −3.87372146782149442686012785408, −3.58510317006896296689691358946, −2.84702398510975616674844560252, −2.67990546288820200222337348371, −1.83323675261394832793354842974, −1.77929909639200479892275302243,
1.77929909639200479892275302243, 1.83323675261394832793354842974, 2.67990546288820200222337348371, 2.84702398510975616674844560252, 3.58510317006896296689691358946, 3.87372146782149442686012785408, 4.61521477311126278475137722645, 5.06059656390981273772252475504, 5.52216342191513768986411681788, 6.11423148148720500758859467564, 6.50910631703067432125052856275, 7.11335892828630119795790234068, 7.57892340876491415450271085909, 8.094738857572422564011297863893, 8.422076705731661692957613300515, 8.589571810335369186252568363816, 9.259740009163592005687544437399, 9.507044163423851554345460825530, 10.14783602097039943396055759284, 10.55348405566603840281525483099
