| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s − 5·7-s + 4·8-s − 2·9-s − 2·11-s − 3·12-s − 9·13-s − 10·14-s + 5·16-s − 3·17-s − 4·18-s + 6·19-s + 5·21-s − 4·22-s − 7·23-s − 4·24-s − 18·26-s + 2·27-s − 15·28-s + 2·29-s − 5·31-s + 6·32-s + 2·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.88·7-s + 1.41·8-s − 2/3·9-s − 0.603·11-s − 0.866·12-s − 2.49·13-s − 2.67·14-s + 5/4·16-s − 0.727·17-s − 0.942·18-s + 1.37·19-s + 1.09·21-s − 0.852·22-s − 1.45·23-s − 0.816·24-s − 3.53·26-s + 0.384·27-s − 2.83·28-s + 0.371·29-s − 0.898·31-s + 1.06·32-s + 0.348·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 85 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 109 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 165 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 107 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 129 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 286 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 207 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.407990081614753518438997617253, −9.349563612736162688935347605120, −8.157589861722829361886774702904, −8.122977075012076541251019641000, −7.52883759681880672876433442106, −6.99003145307253373700181367280, −6.70473469805233267182206547619, −6.56078543410263305001936796313, −5.84236168494095225366956922217, −5.49718281366288437666415243477, −5.22680105345781307133436841878, −4.87870545989729162912470542411, −4.18060447012509745400533690702, −3.78509387676597826168849823887, −3.12032659178808281312473073020, −2.77834456180461400016046101889, −2.50474533305269970826657323526, −1.69465286739611247746851171454, 0, 0,
1.69465286739611247746851171454, 2.50474533305269970826657323526, 2.77834456180461400016046101889, 3.12032659178808281312473073020, 3.78509387676597826168849823887, 4.18060447012509745400533690702, 4.87870545989729162912470542411, 5.22680105345781307133436841878, 5.49718281366288437666415243477, 5.84236168494095225366956922217, 6.56078543410263305001936796313, 6.70473469805233267182206547619, 6.99003145307253373700181367280, 7.52883759681880672876433442106, 8.122977075012076541251019641000, 8.157589861722829361886774702904, 9.349563612736162688935347605120, 9.407990081614753518438997617253
