L(s) = 1 | − 2·2-s + 16·9-s − 16·13-s − 24·17-s − 32·18-s + 10·25-s + 32·26-s − 8·29-s + 32·32-s + 48·34-s + 16·37-s − 112·41-s + 96·49-s − 20·50-s − 176·53-s + 16·58-s + 128·61-s − 64·64-s + 264·73-s − 32·74-s + 50·81-s + 224·82-s − 88·89-s − 264·97-s − 192·98-s + 328·101-s + 352·106-s + ⋯ |
L(s) = 1 | − 2-s + 16/9·9-s − 1.23·13-s − 1.41·17-s − 1.77·18-s + 2/5·25-s + 1.23·26-s − 0.275·29-s + 32-s + 1.41·34-s + 0.432·37-s − 2.73·41-s + 1.95·49-s − 2/5·50-s − 3.32·53-s + 8/29·58-s + 2.09·61-s − 64-s + 3.61·73-s − 0.432·74-s + 0.617·81-s + 2.73·82-s − 0.988·89-s − 2.72·97-s − 1.95·98-s + 3.24·101-s + 3.32·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3426821444\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3426821444\) |
\(L(2)\) |
|
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 | $C_4$ | \( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 3 | $C_2^2:C_4$ | \( 1 - 16 T^{2} + 206 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 96 T^{2} + 6606 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 84 T^{2} - 7674 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 334 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 12 T + 294 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 1124 T^{2} + 571366 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 1856 T^{2} + 1404046 T^{4} - 1856 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 1606 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 2254 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 56 T + 3966 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 6896 T^{2} + 18665806 T^{4} - 6896 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 4736 T^{2} + 14725966 T^{4} - 4736 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 88 T + 7054 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 8164 T^{2} + 34261926 T^{4} - 8164 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 64 T + 5086 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 7536 T^{2} + 47276046 T^{4} - 7536 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 132 T + 9894 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 21296 T^{2} + 201120526 T^{4} - 21296 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 44 T + 8326 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 132 T + 22454 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−13.66048308923934595117485624503, −13.46485836927477789336601076842, −13.08663694101274076841807556316, −12.49875361328906704754609653084, −12.48887786565000405871751767505, −12.21303170706440393411827475958, −11.35638204847778194796798066602, −11.29415588817075777185017400195, −10.88861831861723776689182975920, −10.13985424320236302314452955734, −10.01272837542119754235070824034, −9.788602370359588837916928515174, −9.399943457714330104094633546675, −8.865872522679871094283943224714, −8.592821475085787209603572063168, −8.062354773420145265675760261931, −7.64047041750971698148875835846, −6.98214116978119855428849482800, −6.85329683828389638656960119829, −6.39638741873142387487651468247, −5.43441430703993799926017740945, −4.68474825363585902296547097666, −4.53806032610858065508113104593, −3.53233895242241329213728963533, −2.16603142707187025273094328595,
2.16603142707187025273094328595, 3.53233895242241329213728963533, 4.53806032610858065508113104593, 4.68474825363585902296547097666, 5.43441430703993799926017740945, 6.39638741873142387487651468247, 6.85329683828389638656960119829, 6.98214116978119855428849482800, 7.64047041750971698148875835846, 8.062354773420145265675760261931, 8.592821475085787209603572063168, 8.865872522679871094283943224714, 9.399943457714330104094633546675, 9.788602370359588837916928515174, 10.01272837542119754235070824034, 10.13985424320236302314452955734, 10.88861831861723776689182975920, 11.29415588817075777185017400195, 11.35638204847778194796798066602, 12.21303170706440393411827475958, 12.48887786565000405871751767505, 12.49875361328906704754609653084, 13.08663694101274076841807556316, 13.46485836927477789336601076842, 13.66048308923934595117485624503