| L(s) = 1 | − 6·5-s − 10·7-s + 16·11-s + 26·13-s − 4·17-s − 70·19-s + 128·23-s − 110·25-s + 80·29-s − 250·31-s + 60·35-s − 152·37-s + 146·41-s − 504·43-s + 524·47-s − 498·49-s + 52·53-s − 96·55-s + 164·59-s − 304·61-s − 156·65-s − 914·67-s − 456·73-s − 160·77-s − 824·79-s − 828·83-s + 24·85-s + ⋯ |
| L(s) = 1 | − 0.536·5-s − 0.539·7-s + 0.438·11-s + 0.554·13-s − 0.0570·17-s − 0.845·19-s + 1.16·23-s − 0.879·25-s + 0.512·29-s − 1.44·31-s + 0.289·35-s − 0.675·37-s + 0.556·41-s − 1.78·43-s + 1.62·47-s − 1.45·49-s + 0.134·53-s − 0.235·55-s + 0.361·59-s − 0.638·61-s − 0.297·65-s − 1.66·67-s − 0.731·73-s − 0.236·77-s − 1.17·79-s − 1.09·83-s + 0.0306·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 598 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 16 T + 918 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 70 T + 12118 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 64 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 80 T + 46310 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 250 T + 49782 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 152 T + 70470 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 146 T + 137634 T^{2} - 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 504 T + 215286 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 524 T + 272222 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 52 T + 291198 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 164 T + 406182 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 304 T + 237958 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 914 T + 804838 T^{2} + 914 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 455470 T^{2} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 456 T + 825950 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 824 T + 1009374 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 828 T + 1032470 T^{2} + 828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 826 T + 1571354 T^{2} + 826 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 552 T + 219630 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.273529288320121255977776466755, −9.117213182313598205830608190317, −8.546952968450627424161396078156, −8.441891149763963211912744223343, −7.62675346924794977098064752330, −7.44856267748816354129657924334, −6.79652184929352275694797630984, −6.63124046703839475349939757888, −5.94638755284198231034603410403, −5.70784270031473736785867980071, −4.98717934072757492915860192632, −4.56984230628818185898323009800, −3.82138139149393328989294307802, −3.78069039378975655145513930633, −3.02247893683220023545202022438, −2.55571689606268797681198603124, −1.62009959051762735259982879282, −1.25967319777330728746451238571, 0, 0,
1.25967319777330728746451238571, 1.62009959051762735259982879282, 2.55571689606268797681198603124, 3.02247893683220023545202022438, 3.78069039378975655145513930633, 3.82138139149393328989294307802, 4.56984230628818185898323009800, 4.98717934072757492915860192632, 5.70784270031473736785867980071, 5.94638755284198231034603410403, 6.63124046703839475349939757888, 6.79652184929352275694797630984, 7.44856267748816354129657924334, 7.62675346924794977098064752330, 8.441891149763963211912744223343, 8.546952968450627424161396078156, 9.117213182313598205830608190317, 9.273529288320121255977776466755
