| L(s) = 1 | + 6·3-s − 5·5-s + 27·9-s − 67·11-s − 41·13-s − 30·15-s + 92·17-s + 43·19-s − 148·23-s + 105·25-s + 108·27-s + 77·29-s − 520·31-s − 402·33-s + 7·37-s − 246·39-s + 426·41-s + 107·43-s − 135·45-s + 576·47-s + 552·51-s − 243·53-s + 335·55-s + 258·57-s − 7·59-s − 224·61-s + 205·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 9-s − 1.83·11-s − 0.874·13-s − 0.516·15-s + 1.31·17-s + 0.519·19-s − 1.34·23-s + 0.839·25-s + 0.769·27-s + 0.493·29-s − 3.01·31-s − 2.12·33-s + 0.0311·37-s − 1.01·39-s + 1.62·41-s + 0.379·43-s − 0.447·45-s + 1.78·47-s + 1.51·51-s − 0.629·53-s + 0.821·55-s + 0.599·57-s − 0.0154·59-s − 0.470·61-s + 0.391·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + p T - 16 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 67 T + 3448 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 92 T + 386 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 43 T + 11154 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 148 T + 24430 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 520 T + 125837 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 107 T + 86220 T^{2} - 107 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 576 T + 242170 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 243 T + 309490 T^{2} + 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 200614 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 224 T + 461126 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 687 T + 678832 T^{2} + 687 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 921 T + 578188 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 526 T + 1033727 T^{2} - 526 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 774 T + 966562 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.199057350642650573762978748187, −8.078438837358060382094484112350, −7.58193314722444538517062981921, −7.57678933914051344944378316490, −7.15872249860703764316768439876, −6.74614040972023023945990184685, −5.84051949655477768653289217985, −5.71181474699281042245901602972, −5.27352194504465938286283802917, −4.90505587889398848558486199389, −4.27298837167959193524494303464, −3.90868692286935363576877822366, −3.52881319785242172628897112093, −2.92919899627255661567124204811, −2.52984359047893878765173537965, −2.39653278274862668719268326507, −1.53450241016768092346441125728, −1.07587806306217472055390805781, 0, 0,
1.07587806306217472055390805781, 1.53450241016768092346441125728, 2.39653278274862668719268326507, 2.52984359047893878765173537965, 2.92919899627255661567124204811, 3.52881319785242172628897112093, 3.90868692286935363576877822366, 4.27298837167959193524494303464, 4.90505587889398848558486199389, 5.27352194504465938286283802917, 5.71181474699281042245901602972, 5.84051949655477768653289217985, 6.74614040972023023945990184685, 7.15872249860703764316768439876, 7.57678933914051344944378316490, 7.58193314722444538517062981921, 8.078438837358060382094484112350, 8.199057350642650573762978748187
