| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 6·11-s − 2·13-s − 4·14-s + 5·16-s − 6·17-s − 2·19-s + 12·22-s − 6·23-s − 7·25-s + 4·26-s + 6·28-s − 6·29-s − 2·31-s − 6·32-s + 12·34-s + 4·37-s + 4·38-s − 12·41-s + 4·43-s − 18·44-s + 12·46-s + 6·47-s + 3·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 1.80·11-s − 0.554·13-s − 1.06·14-s + 5/4·16-s − 1.45·17-s − 0.458·19-s + 2.55·22-s − 1.25·23-s − 7/5·25-s + 0.784·26-s + 1.13·28-s − 1.11·29-s − 0.359·31-s − 1.06·32-s + 2.05·34-s + 0.657·37-s + 0.648·38-s − 1.87·41-s + 0.609·43-s − 2.71·44-s + 1.76·46-s + 0.875·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 132 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 247 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−9.419194634971405243524428838064, −9.414361026580805617333864853864, −8.589051670860894876336684020881, −8.488577808480549996085437057326, −7.933903561239261255694151733759, −7.73533071365848616827967978132, −7.32242524797519347987847830371, −6.98157503456671775992331224063, −6.11356301238450005055632412112, −6.10917653893203747419545282984, −5.25303662483731993365747984376, −5.14869922034955435143459742115, −4.19885738821016215672634473444, −4.03484903129711232565482926972, −2.97197453319798944201167887457, −2.52102498828818334383890283128, −1.97740709458130863392849426501, −1.63935128573537334179137429135, 0, 0,
1.63935128573537334179137429135, 1.97740709458130863392849426501, 2.52102498828818334383890283128, 2.97197453319798944201167887457, 4.03484903129711232565482926972, 4.19885738821016215672634473444, 5.14869922034955435143459742115, 5.25303662483731993365747984376, 6.10917653893203747419545282984, 6.11356301238450005055632412112, 6.98157503456671775992331224063, 7.32242524797519347987847830371, 7.73533071365848616827967978132, 7.933903561239261255694151733759, 8.488577808480549996085437057326, 8.589051670860894876336684020881, 9.414361026580805617333864853864, 9.419194634971405243524428838064
