| L(s) = 1 | − 8·2-s − 46·3-s + 32·4-s − 150·5-s + 368·6-s − 494·7-s + 1.05e3·9-s + 1.20e3·10-s + 2.80e3·11-s − 1.47e3·12-s − 5.40e3·13-s + 3.95e3·14-s + 6.90e3·15-s − 1.02e3·16-s + 5.18e3·17-s − 8.46e3·18-s − 4.80e3·20-s + 2.27e4·21-s − 2.24e4·22-s + 4.27e3·23-s + 6.87e3·25-s + 4.32e4·26-s − 3.35e4·27-s − 1.58e4·28-s − 5.52e4·30-s − 7.56e4·31-s + 8.19e3·32-s + ⋯ |
| L(s) = 1 | − 2-s − 1.70·3-s + 1/2·4-s − 6/5·5-s + 1.70·6-s − 1.44·7-s + 1.45·9-s + 6/5·10-s + 2.10·11-s − 0.851·12-s − 2.46·13-s + 1.44·14-s + 2.04·15-s − 1/4·16-s + 1.05·17-s − 1.45·18-s − 3/5·20-s + 2.45·21-s − 2.10·22-s + 0.351·23-s + 0.439·25-s + 2.46·26-s − 1.70·27-s − 0.720·28-s − 2.04·30-s − 2.54·31-s + 1/4·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1679738987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1679738987\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{3} T + p^{5} T^{2} \) |
| 5 | $C_2$ | \( 1 + 6 p^{2} T + p^{6} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 46 T + 1058 T^{2} + 46 p^{6} T^{3} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 494 T + 122018 T^{2} + 494 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 1402 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 5406 T + 14612418 T^{2} + 5406 p^{6} T^{3} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5186 T + 13447298 T^{2} - 5186 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 91133362 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4274 T + 9133538 T^{2} - 4274 p^{6} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 258176242 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 37838 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 74226 T + 2754749538 T^{2} - 74226 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 35438 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78354 T + 3069674658 T^{2} - 78354 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 190386 T + 18123414498 T^{2} - 190386 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 72034 T + 2594448578 T^{2} - 72034 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 83067945682 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 83322 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 121666 T + 7401307778 T^{2} - 121666 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 40318 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 258046 T + 33293869058 T^{2} + 258046 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 210927781442 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 228846 T + 26185245858 T^{2} + 228846 p^{6} T^{3} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 958888783522 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1065666 T + 567822011778 T^{2} - 1065666 p^{6} T^{3} + p^{12} T^{4} \) |
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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
−19.89905126297752912454729502298, −18.97066514379148405079858323693, −18.95270406139479278822887060021, −17.57066677305625424790176005393, −17.05338442944964612721408766329, −16.59774190339917888794105022191, −16.37103147277319680501923015314, −15.14307447814799684737386093760, −14.51473847103721798149995301957, −12.75135291853885834326910510800, −12.06791482750842796238168573781, −11.80748732035280005999843675040, −10.84991820920565446614813894006, −9.680538275033016722032587633368, −9.348948698564673596768843027687, −7.41579948042753569680580278597, −6.96872610397053227181765433980, −5.67199416372430237408373390717, −3.97447928207397225156487965858, −0.45719974308732778039320908703,
0.45719974308732778039320908703, 3.97447928207397225156487965858, 5.67199416372430237408373390717, 6.96872610397053227181765433980, 7.41579948042753569680580278597, 9.348948698564673596768843027687, 9.680538275033016722032587633368, 10.84991820920565446614813894006, 11.80748732035280005999843675040, 12.06791482750842796238168573781, 12.75135291853885834326910510800, 14.51473847103721798149995301957, 15.14307447814799684737386093760, 16.37103147277319680501923015314, 16.59774190339917888794105022191, 17.05338442944964612721408766329, 17.57066677305625424790176005393, 18.95270406139479278822887060021, 18.97066514379148405079858323693, 19.89905126297752912454729502298
