| L(s) = 1 | + 4·4-s + 8·9-s + 7·16-s − 4·19-s + 2·25-s − 28·31-s + 32·36-s + 16·49-s + 12·59-s − 40·61-s + 8·64-s − 12·71-s − 16·76-s + 8·79-s + 33·81-s + 12·89-s + 8·100-s + 36·101-s − 16·109-s − 112·124-s + 127-s + 131-s + 137-s + 139-s + 56·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s + 8/3·9-s + 7/4·16-s − 0.917·19-s + 2/5·25-s − 5.02·31-s + 16/3·36-s + 16/7·49-s + 1.56·59-s − 5.12·61-s + 64-s − 1.42·71-s − 1.83·76-s + 0.900·79-s + 11/3·81-s + 1.27·89-s + 4/5·100-s + 3.58·101-s − 1.53·109-s − 10.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 14/3·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.327297959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.327297959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 31 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 135 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 1266 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4806 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 2439 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 33 p T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 1842 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 40 T^{2} - 2529 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 175 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 304 T^{2} + 40194 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.42820029238289958177891034397, −7.37262725669464713187426925659, −7.26483595409943843274859797967, −7.17444280522297577118819373214, −6.74812415505625711709484517358, −6.67512372258759353124449506476, −6.12413636863721053749206466826, −6.11078711356515679157581910667, −6.03490858953240965662834417194, −5.45130569574316330792314250628, −5.30591327477370796445025117828, −5.08471060381130198038423743851, −4.63029537401670829393468099158, −4.24313808032640236850281893996, −4.11628589610190281891351522260, −4.11624291806191834729918535260, −3.37523115874015794447840729163, −3.30314056029831086999845595764, −3.15590357963151962619131020556, −2.35338357138096273518450029258, −2.21664224541674993110731154087, −1.81202633443824465286063909790, −1.74402302399033843791325800318, −1.43548722526240499851103844497, −0.61272315266525956304406495685,
0.61272315266525956304406495685, 1.43548722526240499851103844497, 1.74402302399033843791325800318, 1.81202633443824465286063909790, 2.21664224541674993110731154087, 2.35338357138096273518450029258, 3.15590357963151962619131020556, 3.30314056029831086999845595764, 3.37523115874015794447840729163, 4.11624291806191834729918535260, 4.11628589610190281891351522260, 4.24313808032640236850281893996, 4.63029537401670829393468099158, 5.08471060381130198038423743851, 5.30591327477370796445025117828, 5.45130569574316330792314250628, 6.03490858953240965662834417194, 6.11078711356515679157581910667, 6.12413636863721053749206466826, 6.67512372258759353124449506476, 6.74812415505625711709484517358, 7.17444280522297577118819373214, 7.26483595409943843274859797967, 7.37262725669464713187426925659, 7.42820029238289958177891034397
