L(s) = 1 | − 4·4-s + 10·9-s + 40·13-s + 12·16-s − 64·19-s + 56·25-s − 128·31-s − 40·36-s + 80·37-s + 80·43-s + 14·49-s − 160·52-s − 56·61-s − 32·64-s + 240·67-s + 120·73-s + 256·76-s − 128·79-s + 19·81-s − 360·97-s − 224·100-s − 160·103-s − 72·109-s + 400·117-s + 272·121-s + 512·124-s + 127-s + ⋯ |
L(s) = 1 | − 4-s + 10/9·9-s + 3.07·13-s + 3/4·16-s − 3.36·19-s + 2.23·25-s − 4.12·31-s − 1.11·36-s + 2.16·37-s + 1.86·43-s + 2/7·49-s − 3.07·52-s − 0.918·61-s − 1/2·64-s + 3.58·67-s + 1.64·73-s + 3.36·76-s − 1.62·79-s + 0.234·81-s − 3.71·97-s − 2.23·100-s − 1.55·103-s − 0.660·109-s + 3.41·117-s + 2.24·121-s + 4.12·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.104426905\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.104426905\) |
\(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 5 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1586 T^{4} - 56 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 20 T + 326 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 440 T^{2} + 204242 T^{4} - 440 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 688 T^{2} + 666818 T^{4} + 688 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 1917638 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 64 T + 2246 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 856 T^{2} - 2330094 T^{4} - 856 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 40 T + 3090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4346 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 28 T + 4838 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 120 T + 12466 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12176 T^{2} + 74167106 T^{4} - 12176 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 60 T + 9766 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 64 T + 10706 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6356 T^{2} - 6983674 T^{4} - 6356 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 27832 T^{2} + 318232338 T^{4} - 27832 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 180 T + 26470 T^{2} + 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−11.80190145350659534624340353683, −11.00940913991854171277319046275, −10.99922140489800898182747560996, −10.97656607736838537880749855765, −10.64329340130072194090857144775, −10.48806283517269266828015213592, −9.497228945272114838058054057530, −9.453655161565392657145870140473, −9.297629870369767403113554207709, −8.618024251648609891486181622236, −8.598351746523521937167876874304, −8.221416607500409401293273300572, −7.984828802260199342245516498849, −7.18679619517840313424465440146, −6.82977106379627389499134696531, −6.67563390324487413403602582116, −5.93492721119400911993870894085, −5.84403132523823845965565988871, −5.31119261484233062466682806038, −4.49681830592379270655001915085, −4.11464777194798800290221936895, −4.00088304407135574630311318648, −3.44301175405416403260060461427, −2.25442640976318514445653685193, −1.23197583297330044693123404601,
1.23197583297330044693123404601, 2.25442640976318514445653685193, 3.44301175405416403260060461427, 4.00088304407135574630311318648, 4.11464777194798800290221936895, 4.49681830592379270655001915085, 5.31119261484233062466682806038, 5.84403132523823845965565988871, 5.93492721119400911993870894085, 6.67563390324487413403602582116, 6.82977106379627389499134696531, 7.18679619517840313424465440146, 7.984828802260199342245516498849, 8.221416607500409401293273300572, 8.598351746523521937167876874304, 8.618024251648609891486181622236, 9.297629870369767403113554207709, 9.453655161565392657145870140473, 9.497228945272114838058054057530, 10.48806283517269266828015213592, 10.64329340130072194090857144775, 10.97656607736838537880749855765, 10.99922140489800898182747560996, 11.00940913991854171277319046275, 11.80190145350659534624340353683