| L(s) = 1 | − 2·4-s + 8·11-s + 3·16-s − 16·19-s + 4·29-s − 12·31-s + 16·41-s − 16·44-s + 8·49-s − 44·59-s + 4·61-s − 4·64-s + 16·71-s + 32·76-s − 8·79-s − 18·81-s − 24·89-s + 44·101-s + 8·109-s − 8·116-s − 4·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 2.41·11-s + 3/4·16-s − 3.67·19-s + 0.742·29-s − 2.15·31-s + 2.49·41-s − 2.41·44-s + 8/7·49-s − 5.72·59-s + 0.512·61-s − 1/2·64-s + 1.89·71-s + 3.67·76-s − 0.900·79-s − 2·81-s − 2.54·89-s + 4.37·101-s + 0.766·109-s − 0.742·116-s − 0.363·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.197996933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197996933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 18 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 7203 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 44 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 138 T^{2} + 9083 T^{4} - 138 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 5378 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 22 T + 233 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 2 T + 117 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 6123 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 104 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6018 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 208 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.71278389644999100581948818429, −6.51261255967793713650543328803, −6.29222070660684011500727322197, −6.25297418006538971067670274510, −5.93906653840011045964252745883, −5.87900272379343897289268249127, −5.38659916503155576644925348815, −5.37360136234552349302611584651, −5.01350637095519203066564162002, −4.41697287223957078050022296075, −4.40907723764914993691765018579, −4.39403923185426901881049264464, −4.27614888635459448993043043867, −3.92776772645798855927164009089, −3.77535928326288717282233463683, −3.41128864377746687933567847092, −3.02016948314528278590651837676, −2.96375911662350427182263275907, −2.45422432814042519338539062846, −2.16217307240048436926906082695, −1.72554745935703660930138230652, −1.62013578213626474716980281381, −1.38491252317642979251150679409, −0.51444890166230282369496532720, −0.49700477550509189015436159945,
0.49700477550509189015436159945, 0.51444890166230282369496532720, 1.38491252317642979251150679409, 1.62013578213626474716980281381, 1.72554745935703660930138230652, 2.16217307240048436926906082695, 2.45422432814042519338539062846, 2.96375911662350427182263275907, 3.02016948314528278590651837676, 3.41128864377746687933567847092, 3.77535928326288717282233463683, 3.92776772645798855927164009089, 4.27614888635459448993043043867, 4.39403923185426901881049264464, 4.40907723764914993691765018579, 4.41697287223957078050022296075, 5.01350637095519203066564162002, 5.37360136234552349302611584651, 5.38659916503155576644925348815, 5.87900272379343897289268249127, 5.93906653840011045964252745883, 6.25297418006538971067670274510, 6.29222070660684011500727322197, 6.51261255967793713650543328803, 6.71278389644999100581948818429
