| L(s) = 1 | + 4·2-s + 3-s + 12·4-s − 5·5-s + 4·6-s + 14·7-s + 25·8-s + 3·9-s − 20·10-s + 4·11-s + 12·12-s − 9·13-s + 56·14-s − 5·15-s + 45·16-s − 17-s + 12·18-s − 9·19-s − 60·20-s + 14·21-s + 16·22-s + 5·23-s + 25·24-s + 10·25-s − 36·26-s + 168·28-s + 5·29-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 6·4-s − 2.23·5-s + 1.63·6-s + 5.29·7-s + 8.83·8-s + 9-s − 6.32·10-s + 1.20·11-s + 3.46·12-s − 2.49·13-s + 14.9·14-s − 1.29·15-s + 45/4·16-s − 0.242·17-s + 2.82·18-s − 2.06·19-s − 13.4·20-s + 3.05·21-s + 3.41·22-s + 1.04·23-s + 5.10·24-s + 2·25-s − 7.06·26-s + 31.7·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.81697108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.81697108\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p^{2} T + p^{2} T^{2} + 7 T^{3} - 21 T^{4} + 7 p T^{5} + p^{4} T^{6} - p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - p T + 25 T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 4 T + 5 T^{2} - 46 T^{3} + 269 T^{4} - 46 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 23 T^{2} + 15 T^{3} + 16 T^{4} + 15 p T^{5} + 23 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 9 T + 17 T^{2} - 3 T^{3} + 100 T^{4} - 3 p T^{5} + 17 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - 5 T - 8 T^{2} - 25 T^{3} + 669 T^{4} - 25 p T^{5} - 8 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 115 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 13 T + 48 T^{2} - 319 T^{3} - 3835 T^{4} - 319 p T^{5} + 48 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 19 T + 204 T^{2} - 1733 T^{3} + 11939 T^{4} - 1733 p T^{5} + 204 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 12 T + 23 T^{2} + 36 T^{3} + 625 T^{4} + 36 p T^{5} + 23 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 4 T - 41 T^{2} - 142 T^{3} + 1599 T^{4} - 142 p T^{5} - 41 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + T - 2 T^{2} - 295 T^{3} + 1491 T^{4} - 295 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 8 T + 55 T^{2} + 178 T^{3} - 981 T^{4} + 178 p T^{5} + 55 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 5 T - 36 T^{2} + 485 T^{3} - 229 T^{4} + 485 p T^{5} - 36 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 32 T + 477 T^{2} - 4660 T^{3} + 38681 T^{4} - 4660 p T^{5} + 477 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 12 T - 17 T^{2} - 666 T^{3} - 3185 T^{4} - 666 p T^{5} - 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 9 T + 8 T^{2} - 585 T^{3} - 5849 T^{4} - 585 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 9 T + 27 T^{2} + 857 T^{3} + 13080 T^{4} + 857 p T^{5} + 27 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 17 T + 2 p T^{2} + 2131 T^{3} + 26289 T^{4} + 2131 p T^{5} + 2 p^{3} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 6 T - 73 T^{2} + 162 T^{3} + 7225 T^{4} + 162 p T^{5} - 73 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 4 T + 69 T^{2} + 68 T^{3} + 3389 T^{4} + 68 p T^{5} + 69 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.956282765369217086522941160195, −7.83513166018730250097943134539, −7.55660703827600231630049844876, −7.42370891199279964747260988454, −7.00395077080954744992830460867, −6.75913636821039501952811659279, −6.74137006495031240695977861753, −6.58859596811431470209221721653, −5.84160848854306944856703071346, −5.39597174991071528412393606003, −5.36188699107157834427828212602, −5.34755731956737623709459662831, −4.86828321957478443403010953717, −4.47685863451723831467101154878, −4.30702943756398238644774227301, −4.20554004691893689590226961450, −4.16491661022648277428382136723, −4.04382901026446601482372566799, −3.18369492812314544036078451150, −2.98284227316966226655030081373, −2.41893618931540249898134203754, −2.38564587474365176566553644526, −1.91424830938483051107285895508, −1.53693611815392051877520411391, −1.38580729821920268617424776165,
1.38580729821920268617424776165, 1.53693611815392051877520411391, 1.91424830938483051107285895508, 2.38564587474365176566553644526, 2.41893618931540249898134203754, 2.98284227316966226655030081373, 3.18369492812314544036078451150, 4.04382901026446601482372566799, 4.16491661022648277428382136723, 4.20554004691893689590226961450, 4.30702943756398238644774227301, 4.47685863451723831467101154878, 4.86828321957478443403010953717, 5.34755731956737623709459662831, 5.36188699107157834427828212602, 5.39597174991071528412393606003, 5.84160848854306944856703071346, 6.58859596811431470209221721653, 6.74137006495031240695977861753, 6.75913636821039501952811659279, 7.00395077080954744992830460867, 7.42370891199279964747260988454, 7.55660703827600231630049844876, 7.83513166018730250097943134539, 7.956282765369217086522941160195
