| L(s) = 1 | + 2-s + 2·3-s − 5-s + 2·6-s + 8-s + 3·9-s − 10-s − 4·11-s + 4·13-s − 2·15-s − 16-s + 3·18-s − 4·22-s + 2·23-s + 2·24-s − 3·25-s + 4·26-s + 10·27-s − 6·29-s − 2·30-s − 6·32-s − 8·33-s − 3·37-s + 8·39-s − 40-s + 6·41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 0.447·5-s + 0.816·6-s + 0.353·8-s + 9-s − 0.316·10-s − 1.20·11-s + 1.10·13-s − 0.516·15-s − 1/4·16-s + 0.707·18-s − 0.852·22-s + 0.417·23-s + 0.408·24-s − 3/5·25-s + 0.784·26-s + 1.92·27-s − 1.11·29-s − 0.365·30-s − 1.06·32-s − 1.39·33-s − 0.493·37-s + 1.28·39-s − 0.158·40-s + 0.937·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.054407147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.054407147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| 1453 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 41 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 156 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−15.6682869625, −15.4231394143, −14.7000824913, −14.3473706293, −13.8782360347, −13.4318003850, −13.1343847341, −12.6270423589, −12.2630763720, −11.2579275331, −10.9284579848, −10.5864073976, −9.70815140933, −9.27014321259, −8.66803752403, −8.10588695796, −7.69648921258, −7.16725255572, −6.36818640928, −5.59982675056, −4.78285456211, −4.31174936423, −3.50568465213, −2.93270094113, −1.82089001253,
1.82089001253, 2.93270094113, 3.50568465213, 4.31174936423, 4.78285456211, 5.59982675056, 6.36818640928, 7.16725255572, 7.69648921258, 8.10588695796, 8.66803752403, 9.27014321259, 9.70815140933, 10.5864073976, 10.9284579848, 11.2579275331, 12.2630763720, 12.6270423589, 13.1343847341, 13.4318003850, 13.8782360347, 14.3473706293, 14.7000824913, 15.4231394143, 15.6682869625
