| L(s) = 1 | + 4·3-s + 8·9-s + 8·13-s − 24·23-s − 2·25-s + 12·27-s + 24·37-s + 32·39-s + 8·47-s + 16·49-s − 16·59-s − 96·69-s − 16·71-s + 40·73-s − 8·75-s + 23·81-s − 8·83-s + 8·97-s − 56·107-s − 48·109-s + 96·111-s + 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 2.21·13-s − 5.00·23-s − 2/5·25-s + 2.30·27-s + 3.94·37-s + 5.12·39-s + 1.16·47-s + 16/7·49-s − 2.08·59-s − 11.5·69-s − 1.89·71-s + 4.68·73-s − 0.923·75-s + 23/9·81-s − 0.878·83-s + 0.812·97-s − 5.41·107-s − 4.59·109-s + 9.11·111-s + 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.836104950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.836104950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 2386 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4082 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_4$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4$ | \( ( 1 - 4 T - 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.67791625374048439321451067749, −6.73000673676486891701030068145, −6.70358510862911210088224607376, −6.68616512729359268528128417691, −6.26216519437737281059893779057, −6.00001014784392624777439472446, −5.97051709847828387592251343797, −5.71450596088105319919339095239, −5.33476764644705104083583278791, −5.30510685353685002044739947340, −4.49469931838036930045635179805, −4.38368658183573462338288319609, −4.27088844762796763341626304012, −3.90962579124419425007846334516, −3.74522112318939901511423993579, −3.66182829044067056772071800503, −3.59988102841603146407170485291, −2.66388554224525805395656689073, −2.60689882874429319990856351568, −2.51780175269678124865062626304, −2.45284117061914577248896877768, −1.73612211265992309580400064701, −1.43274033432367894065856734640, −1.28921250116194665075053671117, −0.47026160017617460907234407445,
0.47026160017617460907234407445, 1.28921250116194665075053671117, 1.43274033432367894065856734640, 1.73612211265992309580400064701, 2.45284117061914577248896877768, 2.51780175269678124865062626304, 2.60689882874429319990856351568, 2.66388554224525805395656689073, 3.59988102841603146407170485291, 3.66182829044067056772071800503, 3.74522112318939901511423993579, 3.90962579124419425007846334516, 4.27088844762796763341626304012, 4.38368658183573462338288319609, 4.49469931838036930045635179805, 5.30510685353685002044739947340, 5.33476764644705104083583278791, 5.71450596088105319919339095239, 5.97051709847828387592251343797, 6.00001014784392624777439472446, 6.26216519437737281059893779057, 6.68616512729359268528128417691, 6.70358510862911210088224607376, 6.73000673676486891701030068145, 7.67791625374048439321451067749
