L(s) = 1 | − 12·3-s − 16·4-s + 16·5-s + 90·9-s + 192·12-s − 192·15-s + 124·16-s − 256·20-s + 136·23-s + 140·25-s − 540·27-s − 204·31-s − 1.44e3·36-s − 172·37-s + 1.44e3·45-s + 664·47-s − 1.48e3·48-s − 198·49-s + 2.52e3·53-s + 360·59-s + 3.07e3·60-s − 896·64-s + 2.62e3·67-s − 1.63e3·69-s + 512·71-s − 1.68e3·75-s + 1.98e3·80-s + ⋯ |
L(s) = 1 | − 2.30·3-s − 2·4-s + 1.43·5-s + 10/3·9-s + 4.61·12-s − 3.30·15-s + 1.93·16-s − 2.86·20-s + 1.23·23-s + 1.11·25-s − 3.84·27-s − 1.18·31-s − 6.66·36-s − 0.764·37-s + 4.77·45-s + 2.06·47-s − 4.47·48-s − 0.577·49-s + 6.53·53-s + 0.794·59-s + 6.60·60-s − 7/4·64-s + 4.79·67-s − 2.84·69-s + 0.855·71-s − 2.58·75-s + 2.77·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4096498840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4096498840\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 11 | | \( 1 \) |
good | 2 | $D_4\times C_2$ | \( 1 + p^{4} T^{2} + 33 p^{2} T^{4} + p^{10} T^{6} + p^{12} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 8 T + 26 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 198 T^{2} + 1691 p^{2} T^{4} + 198 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2644 T^{2} + 9189462 T^{4} + 2644 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9412 T^{2} + 64277574 T^{4} + 9412 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 14310 T^{2} + 108781787 T^{4} + 14310 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 68 T + 16850 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 13508 T^{2} + 921166518 T^{4} - 13508 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 102 T + 50423 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 86 T + 25395 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 41068 T^{2} + 4308998598 T^{4} + 41068 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 284532 T^{2} + 32602404854 T^{4} + 284532 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 332 T + 223442 T^{2} - 332 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 1260 T + 686014 T^{2} - 1260 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 180 T + 410218 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 431934 T^{2} + 130927549811 T^{4} + 431934 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 1314 T + 979175 T^{2} - 1314 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 256 T + 726206 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 754158 T^{2} + 442810688819 T^{4} + 754158 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1133782 T^{2} + 749178836283 T^{4} + 1133782 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2145652 T^{2} + 1800431735574 T^{4} + 2145652 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 272 T + 1341794 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 10 T + 1176411 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.931240906961140293659062423584, −7.31155570181289997032108781731, −7.12910704464939796637996317632, −6.99478935745411240993762609968, −6.86337064921016500647255642636, −6.57484516544314621923020091865, −6.20444136532325799731473317500, −5.75982396687253254435225257412, −5.64281596797749861503070221772, −5.36831464463356971002730958403, −5.31690176904071997174228572022, −5.08517667352010810793173030725, −5.07691184172585268890595284966, −4.32191878309211705815458512916, −4.01793857181663878444104669259, −3.99896586360628751318896996383, −3.91350976194526499336643730823, −3.21943135904682243208796633247, −2.62150211232404210516780471307, −2.14055356919734478072276492639, −2.13258179051229344335552397343, −1.07891989618284122362435970077, −1.07045484945116074236867096800, −0.805615361339239612292655697141, −0.17417325388310127828789612036,
0.17417325388310127828789612036, 0.805615361339239612292655697141, 1.07045484945116074236867096800, 1.07891989618284122362435970077, 2.13258179051229344335552397343, 2.14055356919734478072276492639, 2.62150211232404210516780471307, 3.21943135904682243208796633247, 3.91350976194526499336643730823, 3.99896586360628751318896996383, 4.01793857181663878444104669259, 4.32191878309211705815458512916, 5.07691184172585268890595284966, 5.08517667352010810793173030725, 5.31690176904071997174228572022, 5.36831464463356971002730958403, 5.64281596797749861503070221772, 5.75982396687253254435225257412, 6.20444136532325799731473317500, 6.57484516544314621923020091865, 6.86337064921016500647255642636, 6.99478935745411240993762609968, 7.12910704464939796637996317632, 7.31155570181289997032108781731, 7.931240906961140293659062423584