| L(s) = 1 | + 12·5-s + 20·7-s − 16·11-s − 40·17-s − 40·23-s + 64·25-s + 16·31-s + 240·35-s + 200·41-s + 120·43-s + 200·49-s − 200·53-s − 192·55-s − 312·61-s − 40·67-s − 80·71-s − 20·73-s − 320·77-s − 9·81-s − 240·83-s − 480·85-s + 300·97-s + 232·101-s + 220·103-s − 160·107-s + 160·113-s − 480·115-s + ⋯ |
| L(s) = 1 | + 12/5·5-s + 20/7·7-s − 1.45·11-s − 2.35·17-s − 1.73·23-s + 2.55·25-s + 0.516·31-s + 48/7·35-s + 4.87·41-s + 2.79·43-s + 4.08·49-s − 3.77·53-s − 3.49·55-s − 5.11·61-s − 0.597·67-s − 1.12·71-s − 0.273·73-s − 4.15·77-s − 1/9·81-s − 2.89·83-s − 5.64·85-s + 3.09·97-s + 2.29·101-s + 2.13·103-s − 1.49·107-s + 1.41·113-s − 4.17·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.820733918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.820733918\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 16 p T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1740 T^{3} + 13694 T^{4} - 1740 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 108 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 55678 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 19080 T^{3} + 419714 T^{4} + 19080 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 21126 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 1080 p T^{3} + 1442 p^{2} T^{4} + 1080 p^{3} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2096 T^{2} + 2222466 T^{4} - 2096 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1338 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 3168578 T^{4} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 100 T + 5262 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 432120 T^{3} + 22864898 T^{4} - 432120 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2115554 T^{4} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 200 T + 20000 T^{2} + 1475400 T^{3} + 87973634 T^{4} + 1475400 p^{2} T^{5} + 20000 p^{4} T^{6} + 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5816 T^{2} + 18089586 T^{4} - 5816 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 156 T + 12926 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 170280 T^{3} + 36190274 T^{4} + 170280 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 40 T + 882 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 46140 T^{3} + 1512014 T^{4} + 46140 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22612 T^{2} + 204343398 T^{4} - 22612 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 240 T + 28800 T^{2} + 2549040 T^{3} + 211683458 T^{4} + 2549040 p^{2} T^{5} + 28800 p^{4} T^{6} + 240 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 11716 T^{2} + 151160646 T^{4} + 11716 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 300 T + 45000 T^{2} - 5967300 T^{3} + 681431438 T^{4} - 5967300 p^{2} T^{5} + 45000 p^{4} T^{6} - 300 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−11.10786837151392328385220998428, −10.61940021839009609489343185320, −10.38667621646406219879299925245, −10.38333772758251944639391771482, −9.752001442517049992895065502273, −9.232000355991930346211153698283, −9.221575252487642135957640942607, −8.901707025285830534209501839058, −8.662762912871997279109339698853, −7.88530539513402913080314698975, −7.72911778925758555039472172740, −7.63768400789077808601818569442, −7.46576605683947442746350224208, −6.41310219126849547825542832129, −6.00781343260534877789931637244, −5.95568981056281887558305060510, −5.88571435126036403652192656293, −5.09713493725850525374420201348, −4.56025337077923771172452644817, −4.51166833113304230266642649236, −4.37152550539673925958012979198, −2.69932969823620307452100736355, −2.56332962759671922337603389589, −1.88682070467486011641506462672, −1.61327538530376155463047811506,
1.61327538530376155463047811506, 1.88682070467486011641506462672, 2.56332962759671922337603389589, 2.69932969823620307452100736355, 4.37152550539673925958012979198, 4.51166833113304230266642649236, 4.56025337077923771172452644817, 5.09713493725850525374420201348, 5.88571435126036403652192656293, 5.95568981056281887558305060510, 6.00781343260534877789931637244, 6.41310219126849547825542832129, 7.46576605683947442746350224208, 7.63768400789077808601818569442, 7.72911778925758555039472172740, 7.88530539513402913080314698975, 8.662762912871997279109339698853, 8.901707025285830534209501839058, 9.221575252487642135957640942607, 9.232000355991930346211153698283, 9.752001442517049992895065502273, 10.38333772758251944639391771482, 10.38667621646406219879299925245, 10.61940021839009609489343185320, 11.10786837151392328385220998428
