| L(s) = 1 | + 24·7-s − 124·17-s + 144·23-s + 186·25-s − 408·31-s + 44·41-s + 1.20e3·47-s − 254·49-s − 912·71-s + 1.64e3·73-s − 2.71e3·79-s + 1.87e3·89-s + 2.55e3·97-s + 1.89e3·103-s + 1.24e3·113-s − 2.97e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 3.45e3·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.29·7-s − 1.76·17-s + 1.30·23-s + 1.48·25-s − 2.36·31-s + 0.167·41-s + 3.72·47-s − 0.740·49-s − 1.52·71-s + 2.63·73-s − 3.86·79-s + 2.23·89-s + 2.67·97-s + 1.81·103-s + 1.03·113-s − 2.29·119-s + 1.89·121-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 + 0.000508·157-s + 1.69·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.609544658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.609544658\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 186 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2518 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2054 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 32394 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49322 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 117398 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 600 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 232218 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 274826 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 446906 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 480422 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 456 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 822 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1356 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1131910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 938 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1278 T + p^{3} T^{2} )^{2} \) |
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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
−9.289478690928624504344663680716, −9.177252523231468203718978657844, −8.756373790842805118871184503942, −8.641500388361952812324821211185, −8.011356051206735430009355390670, −7.38924875861704834676787760309, −7.09177982770107387331283098528, −7.07983997740716503887772709393, −6.10023390584808819485700224791, −5.95209827040311355225035888439, −5.14167766176179896237156440009, −4.99247061086554422537695258635, −4.45938509546728866066680399989, −4.12470892235785676545173370293, −3.38408538893782738089383259557, −2.89272935143573295511514797278, −2.04221940129302325136765615199, −1.97665524371833984611449933188, −1.03611510174584248445418526789, −0.51368626996470401308829487096,
0.51368626996470401308829487096, 1.03611510174584248445418526789, 1.97665524371833984611449933188, 2.04221940129302325136765615199, 2.89272935143573295511514797278, 3.38408538893782738089383259557, 4.12470892235785676545173370293, 4.45938509546728866066680399989, 4.99247061086554422537695258635, 5.14167766176179896237156440009, 5.95209827040311355225035888439, 6.10023390584808819485700224791, 7.07983997740716503887772709393, 7.09177982770107387331283098528, 7.38924875861704834676787760309, 8.011356051206735430009355390670, 8.641500388361952812324821211185, 8.756373790842805118871184503942, 9.177252523231468203718978657844, 9.289478690928624504344663680716
