| L(s) = 1 | − 2·4-s + 10·11-s + 3·16-s + 6·19-s − 4·29-s − 8·31-s − 6·41-s − 20·44-s + 49-s − 30·59-s + 20·61-s − 4·64-s − 12·76-s − 32·79-s + 22·89-s + 28·101-s − 20·109-s + 8·116-s + 41·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s + 3.01·11-s + 3/4·16-s + 1.37·19-s − 0.742·29-s − 1.43·31-s − 0.937·41-s − 3.01·44-s + 1/7·49-s − 3.90·59-s + 2.56·61-s − 1/2·64-s − 1.37·76-s − 3.60·79-s + 2.33·89-s + 2.78·101-s − 1.91·109-s + 0.742·116-s + 3.72·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.550269396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550269396\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - T^{2} + 37 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 257 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 542 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 349 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 145 T^{2} + 7993 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 3 T + 83 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 73 T^{2} + 5469 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 205 T^{2} + 16113 T^{4} - 205 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 25342 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 11 T + 107 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 17918 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.48438124361985912078456223765, −6.13652279318468883738037777679, −5.92519771780464827982724088328, −5.89741311582299210726105079259, −5.68178107033425679360631811278, −5.21160398963880597643806605437, −5.12284448288810957373260204322, −5.07948305958241903793738791206, −4.60350068230994527386429882392, −4.45865576331400549084858827504, −4.32924977895421164098362146748, −4.05197536114267903733782690502, −3.74458056479565287756615116183, −3.59844575095653508035875913298, −3.47532061352899991149872684734, −3.32642368734204163368977153249, −2.88987834242495346184355039274, −2.75264965314234214980146324016, −2.21397856768898434198193040191, −1.89264253487393107552143865837, −1.53760872679439923354611822212, −1.49623540516919412202594727365, −1.22730042413765258810942864420, −0.64542249095059494140001186892, −0.41001999777580585966798228766,
0.41001999777580585966798228766, 0.64542249095059494140001186892, 1.22730042413765258810942864420, 1.49623540516919412202594727365, 1.53760872679439923354611822212, 1.89264253487393107552143865837, 2.21397856768898434198193040191, 2.75264965314234214980146324016, 2.88987834242495346184355039274, 3.32642368734204163368977153249, 3.47532061352899991149872684734, 3.59844575095653508035875913298, 3.74458056479565287756615116183, 4.05197536114267903733782690502, 4.32924977895421164098362146748, 4.45865576331400549084858827504, 4.60350068230994527386429882392, 5.07948305958241903793738791206, 5.12284448288810957373260204322, 5.21160398963880597643806605437, 5.68178107033425679360631811278, 5.89741311582299210726105079259, 5.92519771780464827982724088328, 6.13652279318468883738037777679, 6.48438124361985912078456223765
