| L(s) = 1 | − 2·3-s + 4-s − 2·5-s + 2·7-s + 3·9-s + 4·11-s − 2·12-s + 4·15-s − 3·16-s − 4·17-s + 4·19-s − 2·20-s − 4·21-s + 8·23-s + 3·25-s − 4·27-s + 2·28-s − 4·29-s + 12·31-s − 8·33-s − 4·35-s + 3·36-s + 4·37-s − 4·41-s + 4·44-s − 6·45-s + 8·47-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 9-s + 1.20·11-s − 0.577·12-s + 1.03·15-s − 3/4·16-s − 0.970·17-s + 0.917·19-s − 0.447·20-s − 0.872·21-s + 1.66·23-s + 3/5·25-s − 0.769·27-s + 0.377·28-s − 0.742·29-s + 2.15·31-s − 1.39·33-s − 0.676·35-s + 1/2·36-s + 0.657·37-s − 0.624·41-s + 0.603·44-s − 0.894·45-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7853729810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7853729810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−13.83105489238817075034959756976, −13.64300893606418748618108877373, −12.78164098571730908744395799935, −12.21462615368365629827364694348, −11.82381989334064317755500600093, −11.25571416032418356121748741329, −11.22830280592174644561216623642, −10.71248176543795167070725589629, −9.782036460362608195092526135262, −9.200136912201529736656132372640, −8.627168896087875885705621690209, −7.82513692382023103813259447991, −7.22411327079707474917124684286, −6.69481828136884877767437788201, −6.28297263652039351876956159723, −5.25936024831167293412664975803, −4.60463390503738808095240317214, −4.14755026085407614211409726178, −2.89131068890819257465193141069, −1.31435688386558938615415944577,
1.31435688386558938615415944577, 2.89131068890819257465193141069, 4.14755026085407614211409726178, 4.60463390503738808095240317214, 5.25936024831167293412664975803, 6.28297263652039351876956159723, 6.69481828136884877767437788201, 7.22411327079707474917124684286, 7.82513692382023103813259447991, 8.627168896087875885705621690209, 9.200136912201529736656132372640, 9.782036460362608195092526135262, 10.71248176543795167070725589629, 11.22830280592174644561216623642, 11.25571416032418356121748741329, 11.82381989334064317755500600093, 12.21462615368365629827364694348, 12.78164098571730908744395799935, 13.64300893606418748618108877373, 13.83105489238817075034959756976
