| L(s) = 1 | + 3-s − 5·5-s + 6·7-s − 39·9-s + 15·11-s − 59·13-s − 5·15-s − 104·17-s + 38·19-s + 6·21-s + 21·23-s − 217·25-s − 52·27-s − 137·29-s − 4·31-s + 15·33-s − 30·35-s − 152·37-s − 59·39-s − 210·41-s − 67·43-s + 195·45-s − 273·47-s − 431·49-s − 104·51-s + 209·53-s − 75·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.447·5-s + 0.323·7-s − 1.44·9-s + 0.411·11-s − 1.25·13-s − 0.0860·15-s − 1.48·17-s + 0.458·19-s + 0.0623·21-s + 0.190·23-s − 1.73·25-s − 0.370·27-s − 0.877·29-s − 0.0231·31-s + 0.0791·33-s − 0.144·35-s − 0.675·37-s − 0.242·39-s − 0.799·41-s − 0.237·43-s + 0.645·45-s − 0.847·47-s − 1.25·49-s − 0.285·51-s + 0.541·53-s − 0.183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 40 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 242 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 467 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 15 T + 2020 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 59 T + 4566 T^{2} + 59 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 104 T + 11105 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 21 T + 8926 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 137 T + 1804 p T^{2} + 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 48414 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 152 T + 95910 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 210 T + 33442 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 67 T + 152598 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 273 T + 197422 T^{2} + 273 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 209 T + 308546 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 799 T + 513800 T^{2} + 799 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 149 T + 419484 T^{2} - 149 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 p T + 611270 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 792 T + 730138 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 246 T + 719291 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 254 T + 817014 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 374 T + 840590 T^{2} + 374 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 564 T + 942034 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 178 T - 159966 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.15304663806835794752512829985, −10.75689616476067605784796720067, −10.01412763421490167024609217798, −9.606879690117259658554288258623, −9.083091920153420356307397835316, −8.753851119043316632034228063006, −8.004987101059281247638384872180, −7.951092873086802803004512290225, −7.07032596628193750172852107111, −6.85521184108554201507270400521, −5.84096083616091650466185670329, −5.75341720648829651592748817089, −4.74096793102060752049855734440, −4.57524397684749527033976783760, −3.57981196193812879595243040130, −3.13521538248598002552050344488, −2.27307573327774364885560883592, −1.71320616967593342264812056115, 0, 0,
1.71320616967593342264812056115, 2.27307573327774364885560883592, 3.13521538248598002552050344488, 3.57981196193812879595243040130, 4.57524397684749527033976783760, 4.74096793102060752049855734440, 5.75341720648829651592748817089, 5.84096083616091650466185670329, 6.85521184108554201507270400521, 7.07032596628193750172852107111, 7.951092873086802803004512290225, 8.004987101059281247638384872180, 8.753851119043316632034228063006, 9.083091920153420356307397835316, 9.606879690117259658554288258623, 10.01412763421490167024609217798, 10.75689616476067605784796720067, 11.15304663806835794752512829985
