| L(s) = 1 | + 2·3-s − 5-s + 6·7-s + 9-s + 11-s − 10·13-s − 2·15-s + 8·17-s + 5·19-s + 12·21-s + 8·23-s − 4·25-s − 2·27-s − 6·29-s − 2·31-s + 2·33-s − 6·35-s + 3·37-s − 20·39-s + 12·41-s − 14·43-s − 45-s − 12·47-s + 13·49-s + 16·51-s + 11·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 2.26·7-s + 1/3·9-s + 0.301·11-s − 2.77·13-s − 0.516·15-s + 1.94·17-s + 1.14·19-s + 2.61·21-s + 1.66·23-s − 4/5·25-s − 0.384·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s − 1.01·35-s + 0.493·37-s − 3.20·39-s + 1.87·41-s − 2.13·43-s − 0.149·45-s − 1.75·47-s + 13/7·49-s + 2.24·51-s + 1.51·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.368276481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.368276481\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - T - 7 T^{2} + 14 T^{3} - 68 T^{4} + 14 p T^{5} - 7 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 5 T - 5 T^{2} + 40 T^{3} + 64 T^{4} + 40 p T^{5} - 5 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3 T - 53 T^{2} + 36 T^{3} + 2142 T^{4} + 36 p T^{5} - 53 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11 T - T^{2} - 176 T^{3} + 5662 T^{4} - 176 p T^{5} - p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 5 T - 85 T^{2} - 40 T^{3} + 7144 T^{4} - 40 p T^{5} - 85 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 14 p T^{5} - 83 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 14 p T^{5} - 131 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8 T - 53 T^{2} + 328 T^{3} + 2392 T^{4} + 328 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 94 T^{2} + 288 T^{3} + 5775 T^{4} + 288 p T^{5} - 94 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.09624258510186858245766596125, −6.75491187622670982333719145230, −6.70539274080299151270118362105, −6.00441758629331811877208673786, −5.82280641872155081152098719693, −5.78389231917932507122350456380, −5.51406651280828347660093414974, −5.13324664684614581865054366376, −4.96962102694419264657044079771, −4.90192741374399510967025873190, −4.63506943663467239109273843772, −4.60862832622095926368477360840, −4.10367944639262610177503391691, −3.89914073428811462065974805645, −3.40382157830200512312916041621, −3.25838373311175594107893828783, −3.25703942555577172315346299826, −2.98657356086379241540628531753, −2.45182813567123273648155985006, −2.03590285753330902433886161169, −2.01803034621738247170465210646, −1.91061406724528264344558401742, −1.32633112781309284140693355341, −0.73297124190971408099118612339, −0.67830687526878044405244796104,
0.67830687526878044405244796104, 0.73297124190971408099118612339, 1.32633112781309284140693355341, 1.91061406724528264344558401742, 2.01803034621738247170465210646, 2.03590285753330902433886161169, 2.45182813567123273648155985006, 2.98657356086379241540628531753, 3.25703942555577172315346299826, 3.25838373311175594107893828783, 3.40382157830200512312916041621, 3.89914073428811462065974805645, 4.10367944639262610177503391691, 4.60862832622095926368477360840, 4.63506943663467239109273843772, 4.90192741374399510967025873190, 4.96962102694419264657044079771, 5.13324664684614581865054366376, 5.51406651280828347660093414974, 5.78389231917932507122350456380, 5.82280641872155081152098719693, 6.00441758629331811877208673786, 6.70539274080299151270118362105, 6.75491187622670982333719145230, 7.09624258510186858245766596125
