| L(s) = 1 | − 2·3-s − 2·4-s + 5-s − 3·7-s + 2·9-s + 2·11-s + 4·12-s − 2·15-s − 17-s − 9·19-s − 2·20-s + 6·21-s − 7·23-s − 7·25-s − 6·27-s + 6·28-s + 10·29-s − 31-s − 4·33-s − 3·35-s − 4·36-s + 37-s − 41-s + 6·43-s − 4·44-s + 2·45-s − 4·47-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 0.447·5-s − 1.13·7-s + 2/3·9-s + 0.603·11-s + 1.15·12-s − 0.516·15-s − 0.242·17-s − 2.06·19-s − 0.447·20-s + 1.30·21-s − 1.45·23-s − 7/5·25-s − 1.15·27-s + 1.13·28-s + 1.85·29-s − 0.179·31-s − 0.696·33-s − 0.507·35-s − 2/3·36-s + 0.164·37-s − 0.156·41-s + 0.914·43-s − 0.603·44-s + 0.298·45-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10043 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10043 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 9 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 23 T + 239 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 148 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 68 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−16.9966547687, −16.3129190237, −16.0446164812, −15.5161041017, −14.8590649788, −14.1914345022, −13.7952034131, −13.2521326507, −12.8540743465, −12.3899509866, −11.7847040386, −11.3585569059, −10.5035281489, −10.1571628382, −9.61953192145, −9.16424982941, −8.47208911862, −7.85132498437, −6.71186239072, −6.30538217656, −6.00198884623, −5.09439040078, −4.23569988187, −3.85105314039, −2.21429913769, 0,
2.21429913769, 3.85105314039, 4.23569988187, 5.09439040078, 6.00198884623, 6.30538217656, 6.71186239072, 7.85132498437, 8.47208911862, 9.16424982941, 9.61953192145, 10.1571628382, 10.5035281489, 11.3585569059, 11.7847040386, 12.3899509866, 12.8540743465, 13.2521326507, 13.7952034131, 14.1914345022, 14.8590649788, 15.5161041017, 16.0446164812, 16.3129190237, 16.9966547687
