| L(s) = 1 | + 2·7-s − 9-s + 6·11-s − 6·13-s − 6·17-s + 4·19-s + 2·23-s − 6·29-s − 2·37-s + 2·41-s − 6·49-s + 8·53-s − 4·59-s + 4·61-s − 2·63-s − 10·67-s − 20·71-s − 22·73-s + 12·77-s + 4·79-s − 8·81-s − 22·83-s − 12·89-s − 12·91-s − 22·97-s − 6·99-s + 14·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1/3·9-s + 1.80·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.417·23-s − 1.11·29-s − 0.328·37-s + 0.312·41-s − 6/7·49-s + 1.09·53-s − 0.520·59-s + 0.512·61-s − 0.251·63-s − 1.22·67-s − 2.37·71-s − 2.57·73-s + 1.36·77-s + 0.450·79-s − 8/9·81-s − 2.41·83-s − 1.27·89-s − 1.25·91-s − 2.23·97-s − 0.603·99-s + 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 237 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 247 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 270 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−7.38829894664519143692298916959, −7.21752374311486034069924959556, −6.92945069103817562825838584770, −6.70320804630267040149446427924, −6.11033127983421616503409555355, −5.88644197806656755795493532242, −5.36376753330338501982299604204, −5.25322382657969082999794317739, −4.66158948366102977254912488413, −4.34855916059377723299450387272, −4.17560552960840240004853188824, −3.80656096661430630386609935105, −3.10650131664200201660334050632, −2.78127978026704863341866522932, −2.54093778733997818056068738682, −1.81117675267027125945283043481, −1.45621507121329629299942087766, −1.24357861061739028958705129915, 0, 0,
1.24357861061739028958705129915, 1.45621507121329629299942087766, 1.81117675267027125945283043481, 2.54093778733997818056068738682, 2.78127978026704863341866522932, 3.10650131664200201660334050632, 3.80656096661430630386609935105, 4.17560552960840240004853188824, 4.34855916059377723299450387272, 4.66158948366102977254912488413, 5.25322382657969082999794317739, 5.36376753330338501982299604204, 5.88644197806656755795493532242, 6.11033127983421616503409555355, 6.70320804630267040149446427924, 6.92945069103817562825838584770, 7.21752374311486034069924959556, 7.38829894664519143692298916959
