| L(s) = 1 | + 2-s + 2·3-s + 4-s − 3·5-s + 2·6-s − 3·7-s + 3·8-s + 3·9-s − 3·10-s − 4·11-s + 2·12-s − 3·14-s − 6·15-s + 16-s + 17-s + 3·18-s + 6·19-s − 3·20-s − 6·21-s − 4·22-s + 4·23-s + 6·24-s + 25-s + 4·27-s − 3·28-s + 29-s − 6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 9-s − 0.948·10-s − 1.20·11-s + 0.577·12-s − 0.801·14-s − 1.54·15-s + 1/4·16-s + 0.242·17-s + 0.707·18-s + 1.37·19-s − 0.670·20-s − 1.30·21-s − 0.852·22-s + 0.834·23-s + 1.22·24-s + 1/5·25-s + 0.769·27-s − 0.566·28-s + 0.185·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.902178037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.902178037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 98 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 136 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 165 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 232 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−11.15065115819270685995271881214, −10.82589628888930509222156347347, −10.14393115742285463261574157805, −9.849230605811608010124469113557, −9.341350995920564970294587835178, −8.996913759646670549518095881824, −8.091320633891752412812506182646, −7.85652805933943892114261188810, −7.77503323002833520594102119833, −6.93247894334291483124128575130, −6.92326777771850493959671438728, −6.06535587820635341798754774064, −5.25913165509084156656134422835, −4.98743209888744746823088102972, −4.08495378148245155731356427379, −3.95317298693862944697027814067, −3.21907311673498815231597021811, −2.83432581445068102501033746268, −2.24384209720259242031140313069, −0.877085449946524228231774992588,
0.877085449946524228231774992588, 2.24384209720259242031140313069, 2.83432581445068102501033746268, 3.21907311673498815231597021811, 3.95317298693862944697027814067, 4.08495378148245155731356427379, 4.98743209888744746823088102972, 5.25913165509084156656134422835, 6.06535587820635341798754774064, 6.92326777771850493959671438728, 6.93247894334291483124128575130, 7.77503323002833520594102119833, 7.85652805933943892114261188810, 8.091320633891752412812506182646, 8.996913759646670549518095881824, 9.341350995920564970294587835178, 9.849230605811608010124469113557, 10.14393115742285463261574157805, 10.82589628888930509222156347347, 11.15065115819270685995271881214
