| L(s) = 1 | − 4·4-s + 6·7-s − 9-s − 6·13-s + 12·16-s − 24·28-s + 12·29-s + 4·36-s − 18·37-s − 24·47-s + 13·49-s + 24·52-s + 2·61-s − 6·63-s − 32·64-s − 12·73-s − 2·79-s + 81-s + 12·83-s − 36·91-s − 18·97-s + 36·101-s + 72·112-s − 48·116-s + 6·117-s + 13·121-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.26·7-s − 1/3·9-s − 1.66·13-s + 3·16-s − 4.53·28-s + 2.22·29-s + 2/3·36-s − 2.95·37-s − 3.50·47-s + 13/7·49-s + 3.32·52-s + 0.256·61-s − 0.755·63-s − 4·64-s − 1.40·73-s − 0.225·79-s + 1/9·81-s + 1.31·83-s − 3.77·91-s − 1.82·97-s + 3.58·101-s + 6.80·112-s − 4.45·116-s + 0.554·117-s + 1.18·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.027982543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027982543\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
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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
−10.16178093561568207411528352886, −9.621242986385410522860354654034, −9.572753081722766559027705894158, −8.684657648143539105925243965412, −8.541959166470814211997922448245, −8.219797624410430169766163816202, −8.068162760093346448063144212964, −7.35670306004167929745029316726, −7.05391899394968082644248293201, −6.30035967152762208445972269282, −5.60335769776578956065934969318, −5.13794926523942514939117518867, −4.86817028255218284289179759277, −4.67017223267876374464878299154, −4.37172509468077663839188070438, −3.35563664710715014783271783726, −3.13815857071687185718802727582, −1.96368688157241976088929021790, −1.54568013163772506708277889737, −0.50507812086146307515836571268,
0.50507812086146307515836571268, 1.54568013163772506708277889737, 1.96368688157241976088929021790, 3.13815857071687185718802727582, 3.35563664710715014783271783726, 4.37172509468077663839188070438, 4.67017223267876374464878299154, 4.86817028255218284289179759277, 5.13794926523942514939117518867, 5.60335769776578956065934969318, 6.30035967152762208445972269282, 7.05391899394968082644248293201, 7.35670306004167929745029316726, 8.068162760093346448063144212964, 8.219797624410430169766163816202, 8.541959166470814211997922448245, 8.684657648143539105925243965412, 9.572753081722766559027705894158, 9.621242986385410522860354654034, 10.16178093561568207411528352886
