| L(s) = 1 | − 18·9-s + 36·11-s + 148·19-s − 192·29-s − 16·31-s − 672·41-s − 98·49-s − 852·59-s − 1.43e3·61-s + 1.33e3·71-s − 2.21e3·79-s + 243·81-s − 4.09e3·89-s − 648·99-s − 660·101-s − 3.40e3·109-s − 2.42e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.64e3·169-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 0.986·11-s + 1.78·19-s − 1.22·29-s − 0.0926·31-s − 2.55·41-s − 2/7·49-s − 1.88·59-s − 3.00·61-s + 2.22·71-s − 3.15·79-s + 1/3·81-s − 4.87·89-s − 0.657·99-s − 0.650·101-s − 2.99·109-s − 1.82·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.746·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.782050067\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.782050067\times10^{-5}\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 11 | $D_{4}$ | \( ( 1 - 18 T + 1699 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1640 T^{2} - 1184082 T^{4} - 1640 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 15080 T^{2} + 103012638 T^{4} - 15080 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 74 T + 14826 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 32378 T^{2} + 929787 p^{2} T^{4} - 32378 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 96 T + 19501 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T - 24966 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 145994 T^{2} + 10229395227 T^{4} - 145994 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 336 T + 99250 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 291650 T^{2} + 33887194323 T^{4} - 291650 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 359888 T^{2} + 53383449390 T^{4} - 359888 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 19148 T^{2} + 33942565878 T^{4} - 19148 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 426 T + 98818 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 716 T + 581082 T^{2} + 716 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 839642 T^{2} + 325101266763 T^{4} - 839642 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 666 T + 825667 T^{2} - 666 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 591704 T^{2} + 158076068958 T^{4} - 591704 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1108 T + 1198773 T^{2} + 1108 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2233508 T^{2} + 1900365379638 T^{4} - 2233508 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 2046 T + 2205646 T^{2} + 2046 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 3239996 T^{2} + 4289484356358 T^{4} - 3239996 p^{6} T^{6} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.01666607358780225545454613737, −5.89154375818139691646728211550, −5.74561599664743000808591960773, −5.30366652756118233957320857270, −5.27500787010155833885705168828, −5.24116999239435761935376716916, −4.95264841102573414672517813374, −4.47824437488054513302019601541, −4.34969860569445776721896298616, −4.22625109935107342877400395872, −3.99392359509846121503866651184, −3.58894061375998179856502962173, −3.44279193806723979481296908110, −3.16651947867081661620185571699, −3.10003564166512332871328701670, −2.88404001857655129864799518714, −2.49046779439647257461906318482, −2.18289925873317309977539831231, −2.03581866567981563757891311737, −1.36741397073879450705050185770, −1.27225162223448056103264542995, −1.23279506292524447682194074727, −1.23051524196182922337574245514, −0.05350472270403746864646183934, −0.00488301854964920118318691228,
0.00488301854964920118318691228, 0.05350472270403746864646183934, 1.23051524196182922337574245514, 1.23279506292524447682194074727, 1.27225162223448056103264542995, 1.36741397073879450705050185770, 2.03581866567981563757891311737, 2.18289925873317309977539831231, 2.49046779439647257461906318482, 2.88404001857655129864799518714, 3.10003564166512332871328701670, 3.16651947867081661620185571699, 3.44279193806723979481296908110, 3.58894061375998179856502962173, 3.99392359509846121503866651184, 4.22625109935107342877400395872, 4.34969860569445776721896298616, 4.47824437488054513302019601541, 4.95264841102573414672517813374, 5.24116999239435761935376716916, 5.27500787010155833885705168828, 5.30366652756118233957320857270, 5.74561599664743000808591960773, 5.89154375818139691646728211550, 6.01666607358780225545454613737
