| L(s) = 1 | − 2-s + 3-s − 2·4-s + 5-s − 6-s + 3·8-s − 4·9-s − 10-s − 5·11-s − 2·12-s + 4·13-s + 15-s + 16-s − 11·17-s + 4·18-s − 3·19-s − 2·20-s + 5·22-s + 2·23-s + 3·24-s + 2·25-s − 4·26-s − 6·27-s − 30-s − 9·31-s − 2·32-s − 5·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s − 4/3·9-s − 0.316·10-s − 1.50·11-s − 0.577·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s − 2.66·17-s + 0.942·18-s − 0.688·19-s − 0.447·20-s + 1.06·22-s + 0.417·23-s + 0.612·24-s + 2/5·25-s − 0.784·26-s − 1.15·27-s − 0.182·30-s − 1.61·31-s − 0.353·32-s − 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 707281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 707281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 29 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 71 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - T + 87 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 127 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 137 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.711378688188144839945369600518, −9.453766446592641792838607570399, −9.097691096245418506206456907826, −8.583496549012541642589046674846, −8.405454958464993032384712060409, −8.318697189967849552966517337284, −7.74168767004205784245139163213, −6.83918953864746637229187488986, −6.72512851243885830412485772629, −6.03699646126926180927547171328, −5.50562936652875576665194517298, −5.11913862042912634519027874535, −4.56508001977196221481393428336, −4.19535437260133487445131990904, −3.18543640135487002394635092902, −3.10535176000416497470360599770, −2.13944074327352220016671955315, −1.73987673421271270695939834898, 0, 0,
1.73987673421271270695939834898, 2.13944074327352220016671955315, 3.10535176000416497470360599770, 3.18543640135487002394635092902, 4.19535437260133487445131990904, 4.56508001977196221481393428336, 5.11913862042912634519027874535, 5.50562936652875576665194517298, 6.03699646126926180927547171328, 6.72512851243885830412485772629, 6.83918953864746637229187488986, 7.74168767004205784245139163213, 8.318697189967849552966517337284, 8.405454958464993032384712060409, 8.583496549012541642589046674846, 9.097691096245418506206456907826, 9.453766446592641792838607570399, 9.711378688188144839945369600518
