| L(s) = 1 | + 8·5-s + 10·7-s + 5·9-s + 20·11-s + 30·17-s + 12·19-s − 70·23-s − 2·25-s + 80·35-s + 40·43-s + 40·45-s − 20·47-s − 23·49-s + 160·55-s − 80·61-s + 50·63-s + 210·73-s + 200·77-s − 56·81-s + 80·83-s + 240·85-s + 96·95-s + 100·99-s − 100·101-s − 560·115-s + 300·119-s + 58·121-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 10/7·7-s + 5/9·9-s + 1.81·11-s + 1.76·17-s + 0.631·19-s − 3.04·23-s − 0.0799·25-s + 16/7·35-s + 0.930·43-s + 8/9·45-s − 0.425·47-s − 0.469·49-s + 2.90·55-s − 1.31·61-s + 0.793·63-s + 2.87·73-s + 2.59·77-s − 0.691·81-s + 0.963·83-s + 2.82·85-s + 1.01·95-s + 1.01·99-s − 0.990·101-s − 4.86·115-s + 2.52·119-s + 0.479·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.434964460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.434964460\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 12 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 p T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1357 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2270 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 115 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6637 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7405 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )( 1 + 92 T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 105 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3790 T^{2} + p^{4} T^{4} \) |
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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
−11.71236281347349385293397596828, −11.49072953369392320416378780383, −10.62754449133093429769075492058, −10.27642103371244524356254942221, −9.715359653116220709085604869621, −9.502355967712997368616536105475, −9.229411402009606104586401020741, −8.117644721541530602657845259196, −8.042045177349578926682590277781, −7.59125545133007173181906500571, −6.66159395655224827275762307886, −6.28427713561947794511190600938, −5.68581313284993495236243787399, −5.47190529493409477537670994970, −4.59440961426593279636640445576, −4.01801244556623368013018356485, −3.47867546913580386536958382977, −2.16291611379945237585452437211, −1.69318189688117395294782880891, −1.23838962591098929658234863504,
1.23838962591098929658234863504, 1.69318189688117395294782880891, 2.16291611379945237585452437211, 3.47867546913580386536958382977, 4.01801244556623368013018356485, 4.59440961426593279636640445576, 5.47190529493409477537670994970, 5.68581313284993495236243787399, 6.28427713561947794511190600938, 6.66159395655224827275762307886, 7.59125545133007173181906500571, 8.042045177349578926682590277781, 8.117644721541530602657845259196, 9.229411402009606104586401020741, 9.502355967712997368616536105475, 9.715359653116220709085604869621, 10.27642103371244524356254942221, 10.62754449133093429769075492058, 11.49072953369392320416378780383, 11.71236281347349385293397596828
