| L(s) = 1 | − 32·7-s − 10·9-s − 60·17-s − 96·23-s + 150·25-s + 640·31-s − 820·41-s − 832·47-s + 82·49-s + 320·63-s − 800·71-s + 1.26e3·73-s − 2.24e3·79-s − 629·81-s + 652·89-s − 220·97-s + 96·103-s + 2.98e3·113-s + 1.92e3·119-s + 1.06e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 600·153-s + ⋯ |
| L(s) = 1 | − 1.72·7-s − 0.370·9-s − 0.856·17-s − 0.870·23-s + 6/5·25-s + 3.70·31-s − 3.12·41-s − 2.58·47-s + 0.239·49-s + 0.639·63-s − 1.33·71-s + 2.02·73-s − 3.19·79-s − 0.862·81-s + 0.776·89-s − 0.230·97-s + 0.0918·103-s + 2.48·113-s + 1.47·119-s + 0.797·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.317·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6788022850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6788022850\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )( 1 + 4 p T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1062 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1894 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12118 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 47622 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 320 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 5206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 135910 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 416 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 129654 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 370758 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 453062 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 650 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 400 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1120 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 838870 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 326 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{2} \) |
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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
−12.01474191377972199804517908737, −11.44536226817632417900968605724, −10.83324215545000384587872783832, −10.08748715585605361330412484678, −9.860324009591381400190213278541, −9.797322519498250755319560144256, −8.719162236737348927989245241226, −8.499641150113475264753038129526, −8.148337243795720213392314545155, −7.14087511100931318256000810300, −6.67998484859997601293902990343, −6.34847506714759986971708148501, −6.03631809935367434881268009936, −4.80897248452884833353152636402, −4.79297529878926784415437403741, −3.68933057373185752146135733512, −3.06399619707349571550346348419, −2.72445616511619237991975855785, −1.54672260690973603583557748867, −0.32173100418443105585881673192,
0.32173100418443105585881673192, 1.54672260690973603583557748867, 2.72445616511619237991975855785, 3.06399619707349571550346348419, 3.68933057373185752146135733512, 4.79297529878926784415437403741, 4.80897248452884833353152636402, 6.03631809935367434881268009936, 6.34847506714759986971708148501, 6.67998484859997601293902990343, 7.14087511100931318256000810300, 8.148337243795720213392314545155, 8.499641150113475264753038129526, 8.719162236737348927989245241226, 9.797322519498250755319560144256, 9.860324009591381400190213278541, 10.08748715585605361330412484678, 10.83324215545000384587872783832, 11.44536226817632417900968605724, 12.01474191377972199804517908737
