| L(s) = 1 | − 4·2-s + 12·4-s − 7·5-s − 32·8-s + 28·10-s − 25·11-s + 49·13-s + 80·16-s − 98·17-s + 119·19-s − 84·20-s + 100·22-s + 122·23-s − 165·25-s − 196·26-s + 73·29-s − 98·31-s − 192·32-s + 392·34-s + 289·37-s − 476·38-s + 224·40-s − 336·41-s + 307·43-s − 300·44-s − 488·46-s − 672·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.626·5-s − 1.41·8-s + 0.885·10-s − 0.685·11-s + 1.04·13-s + 5/4·16-s − 1.39·17-s + 1.43·19-s − 0.939·20-s + 0.969·22-s + 1.10·23-s − 1.31·25-s − 1.47·26-s + 0.467·29-s − 0.567·31-s − 1.06·32-s + 1.97·34-s + 1.28·37-s − 2.03·38-s + 0.885·40-s − 1.27·41-s + 1.08·43-s − 1.02·44-s − 1.56·46-s − 2.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 7 T + 214 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 25 T + 454 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 49 T + 3788 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 98 T + 7402 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 119 T + 16824 T^{2} - 119 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 122 T + 18598 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 73 T + 47746 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 98 T + 24155 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 289 T + 100908 T^{2} - 289 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 672 T + 292750 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 375 T + 141406 T^{2} + 375 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 763 T + 376762 T^{2} + 763 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 406 T + 439394 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1041 T + 813340 T^{2} + 1041 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1652 T + 1360270 T^{2} - 1652 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 189 T + 332980 T^{2} - 189 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 524 T + 591329 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 287 T + 781978 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2394 T + 2702050 T^{2} + 2394 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 63 T + 667132 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.387467350183254222444125160846, −9.253090624008808895699411053299, −8.612704906059426280862084524540, −8.350336307876207459631768824872, −7.77240553284491167487940665479, −7.73482935857194203677128015887, −7.09957180251424029552736089372, −6.69347659385950267891146737612, −6.18476559577194290316695853626, −5.86610041828587300779013465969, −4.96672594782918422361376718589, −4.87574529762245979549440910788, −3.79430313647371614291145859694, −3.61382417421161474660633470060, −2.78010997241749435217309452774, −2.45462485663241713426549120986, −1.44501379208625896681188716243, −1.19454548994668685365480765120, 0, 0,
1.19454548994668685365480765120, 1.44501379208625896681188716243, 2.45462485663241713426549120986, 2.78010997241749435217309452774, 3.61382417421161474660633470060, 3.79430313647371614291145859694, 4.87574529762245979549440910788, 4.96672594782918422361376718589, 5.86610041828587300779013465969, 6.18476559577194290316695853626, 6.69347659385950267891146737612, 7.09957180251424029552736089372, 7.73482935857194203677128015887, 7.77240553284491167487940665479, 8.350336307876207459631768824872, 8.612704906059426280862084524540, 9.253090624008808895699411053299, 9.387467350183254222444125160846
