| L(s) = 1 | − 4·3-s + 2·4-s − 5·5-s − 12·7-s + 13·9-s + 4·11-s − 8·12-s + 13-s + 20·15-s + 4·17-s + 17·19-s − 10·20-s + 48·21-s + 18·23-s + 10·25-s − 30·27-s − 24·28-s − 29-s + 9·31-s − 16·33-s + 60·35-s + 26·36-s + 22·37-s − 4·39-s − 2·41-s + 8·44-s − 65·45-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4-s − 2.23·5-s − 4.53·7-s + 13/3·9-s + 1.20·11-s − 2.30·12-s + 0.277·13-s + 5.16·15-s + 0.970·17-s + 3.90·19-s − 2.23·20-s + 10.4·21-s + 3.75·23-s + 2·25-s − 5.77·27-s − 4.53·28-s − 0.185·29-s + 1.61·31-s − 2.78·33-s + 10.1·35-s + 13/3·36-s + 3.61·37-s − 0.640·39-s − 0.312·41-s + 1.20·44-s − 9.68·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9855310958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9855310958\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 4 T + p T^{2} - 10 T^{3} - 29 T^{4} - 10 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 4 T - 5 T^{2} + 34 T^{3} - 21 T^{4} + 34 p T^{5} - 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 25 p T^{5} - 12 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 4 T - T^{2} - 58 T^{3} + 509 T^{4} - 58 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 17 T + 165 T^{2} - 1117 T^{3} + 5564 T^{4} - 1117 p T^{5} + 165 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 18 T + 7 p T^{2} - 1014 T^{3} + 5239 T^{4} - 1014 p T^{5} + 7 p^{3} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 9 T + 75 T^{2} - 521 T^{3} + 3864 T^{4} - 521 p T^{5} + 75 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 22 T + 207 T^{2} - 1220 T^{3} + 6701 T^{4} - 1220 p T^{5} + 207 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 2 T - 37 T^{2} + 44 T^{3} + 1805 T^{4} + 44 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 6 T - 11 T^{2} - 222 T^{3} + 3559 T^{4} - 222 p T^{5} - 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 12 T + 91 T^{2} + 966 T^{3} + 9829 T^{4} + 966 p T^{5} + 91 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 5 T + 56 T^{2} - 245 T^{3} + 1001 T^{4} - 245 p T^{5} + 56 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 25 T + 399 T^{2} - 4655 T^{3} + 41216 T^{4} - 4655 p T^{5} + 399 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 5 T + 48 T^{2} - 85 T^{3} + 449 T^{4} - 85 p T^{5} + 48 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 19 T + 115 T^{2} - 731 T^{3} + 8664 T^{4} - 731 p T^{5} + 115 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 12 T - 19 T^{2} - 684 T^{3} - 3131 T^{4} - 684 p T^{5} - 19 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 15 T + 186 T^{2} + 2165 T^{3} + 25461 T^{4} + 2165 p T^{5} + 186 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 765 p T^{5} - 2 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 200 p T^{5} - 29 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 + 3 T - 88 T^{2} - 555 T^{3} + 6871 T^{4} - 555 p T^{5} - 88 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.25205001851517198163923993987, −7.09400605972527105923193635777, −6.91270610900129720090057230918, −6.87181476351277284256598020079, −6.66862836013684568631951857300, −6.31722864979401774469825094270, −6.20537967375414099546141759548, −5.99750780871024952974507020583, −5.71695210883755235224885142255, −5.48414169242751498864506069527, −4.97626632457174106922633021880, −4.86656483889818087519935527578, −4.77192481651649286842720702441, −4.15345296422631776755428294962, −3.93911531354499104330515039006, −3.73734372212636285315018150093, −3.45033329069792572344723920105, −3.20876741440552703121032585407, −3.14308043286597520653668470191, −2.71097861620314779314619881340, −2.60426110922958210372188461347, −1.27048048163683067337447150366, −1.03068765367598748626797868307, −0.792494258648040294782538604505, −0.59364550121736750999275440248,
0.59364550121736750999275440248, 0.792494258648040294782538604505, 1.03068765367598748626797868307, 1.27048048163683067337447150366, 2.60426110922958210372188461347, 2.71097861620314779314619881340, 3.14308043286597520653668470191, 3.20876741440552703121032585407, 3.45033329069792572344723920105, 3.73734372212636285315018150093, 3.93911531354499104330515039006, 4.15345296422631776755428294962, 4.77192481651649286842720702441, 4.86656483889818087519935527578, 4.97626632457174106922633021880, 5.48414169242751498864506069527, 5.71695210883755235224885142255, 5.99750780871024952974507020583, 6.20537967375414099546141759548, 6.31722864979401774469825094270, 6.66862836013684568631951857300, 6.87181476351277284256598020079, 6.91270610900129720090057230918, 7.09400605972527105923193635777, 7.25205001851517198163923993987
