| L(s) = 1 | + 2-s − 3-s + 5·5-s − 6-s + 7-s + 5·10-s + 11-s − 6·13-s + 14-s − 5·15-s − 4·17-s − 11·19-s − 21-s + 22-s − 6·23-s + 20·25-s − 6·26-s + 4·29-s − 5·30-s + 17·31-s − 32-s − 33-s − 4·34-s + 5·35-s + 10·37-s − 11·38-s + 6·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 2.23·5-s − 0.408·6-s + 0.377·7-s + 1.58·10-s + 0.301·11-s − 1.66·13-s + 0.267·14-s − 1.29·15-s − 0.970·17-s − 2.52·19-s − 0.218·21-s + 0.213·22-s − 1.25·23-s + 4·25-s − 1.17·26-s + 0.742·29-s − 0.912·30-s + 3.05·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-s + 0.845·35-s + 1.64·37-s − 1.78·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.120438867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.120438867\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2$$\times$$C_4$ | \( ( 1 - p T^{2} )^{2}( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2:C_4$ | \( 1 + 6 T + 3 T^{2} - 10 T^{3} + 81 T^{4} - 10 p T^{5} + 3 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 4 T + 29 T^{2} + 148 T^{3} + 389 T^{4} + 148 p T^{5} + 29 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 11 T + 77 T^{2} + 463 T^{3} + 2380 T^{4} + 463 p T^{5} + 77 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 4 T - 13 T^{2} - 82 T^{3} + 1205 T^{4} - 82 p T^{5} - 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 17 T + 83 T^{2} + 191 T^{3} - 3320 T^{4} + 191 p T^{5} + 83 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 10 T + 23 T^{2} + 280 T^{3} - 3191 T^{4} + 280 p T^{5} + 23 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 16 T + 65 T^{2} - 636 T^{3} - 8251 T^{4} - 636 p T^{5} + 65 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 24 T + 7 p T^{2} + 3318 T^{3} + 25669 T^{4} + 3318 p T^{5} + 7 p^{3} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - 20 T + 187 T^{2} - 1600 T^{3} + 13449 T^{4} - 1600 p T^{5} + 187 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 20 T + 181 T^{2} - 1600 T^{3} + 14601 T^{4} - 1600 p T^{5} + 181 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 2 T + 3 T^{2} + 326 T^{3} + 1265 T^{4} + 326 p T^{5} + 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 16 T + 25 T^{2} - 916 T^{3} - 9471 T^{4} - 916 p T^{5} + 25 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 8 T - 9 T^{2} + 656 T^{3} - 4591 T^{4} + 656 p T^{5} - 9 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 14 T + 57 T^{2} + 862 T^{3} + 13865 T^{4} + 862 p T^{5} + 57 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 8 T + 101 T^{2} + 794 T^{3} + 2909 T^{4} + 794 p T^{5} + 101 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 17 T + 239 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 4 T - T^{2} - 682 T^{3} + 829 T^{4} - 682 p T^{5} - p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.076937112992542481300009992854, −7.931894702525411979018098363031, −7.15247380164770849142465848160, −7.09070517651715862638362293703, −6.90824238562105365446727109219, −6.60946481345773416199663192588, −6.59141324374096121560048045181, −6.12100220900488345936825434537, −5.97117666017321595232030540762, −5.86265304610921556627310879370, −5.50528416350705273376961950473, −5.16929342702029011299789106407, −4.87194595095778443472059548526, −4.71905236663916882994324968400, −4.47084660319107994613558550203, −4.32578690140140855423390724882, −4.02556044693945944124665351080, −3.53505822108427828487962040709, −2.81975941803710514967787250089, −2.72144395257292170215943227090, −2.37534112523506956917378762663, −2.29681415650743962909510735681, −1.71994509486623232268839802055, −1.38746255571884355257263658326, −0.54236191123585948230760585494,
0.54236191123585948230760585494, 1.38746255571884355257263658326, 1.71994509486623232268839802055, 2.29681415650743962909510735681, 2.37534112523506956917378762663, 2.72144395257292170215943227090, 2.81975941803710514967787250089, 3.53505822108427828487962040709, 4.02556044693945944124665351080, 4.32578690140140855423390724882, 4.47084660319107994613558550203, 4.71905236663916882994324968400, 4.87194595095778443472059548526, 5.16929342702029011299789106407, 5.50528416350705273376961950473, 5.86265304610921556627310879370, 5.97117666017321595232030540762, 6.12100220900488345936825434537, 6.59141324374096121560048045181, 6.60946481345773416199663192588, 6.90824238562105365446727109219, 7.09070517651715862638362293703, 7.15247380164770849142465848160, 7.931894702525411979018098363031, 8.076937112992542481300009992854
