| L(s) = 1 | + 2·3-s + 4-s + 2·7-s + 9-s + 8·11-s + 2·12-s − 12·17-s + 4·21-s + 25-s − 2·27-s + 2·28-s + 16·33-s + 36-s − 2·41-s + 8·44-s + 12·47-s + 3·49-s − 24·51-s + 2·53-s − 18·59-s + 6·61-s + 2·63-s − 64-s − 12·68-s + 12·71-s + 44·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 2.41·11-s + 0.577·12-s − 2.91·17-s + 0.872·21-s + 1/5·25-s − 0.384·27-s + 0.377·28-s + 2.78·33-s + 1/6·36-s − 0.312·41-s + 1.20·44-s + 1.75·47-s + 3/7·49-s − 3.36·51-s + 0.274·53-s − 2.34·59-s + 0.768·61-s + 0.251·63-s − 1/8·64-s − 1.45·68-s + 1.42·71-s + 5.14·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.922026236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.922026236\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 22 T^{3} - 68 T^{4} + 22 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 90 T^{2} + 504 T^{3} + 2291 T^{4} + 504 p T^{5} + 90 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 29 T^{2} + 480 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2106 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 102 T^{2} + 4235 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 2 T - 52 T^{2} - 52 T^{3} + 1291 T^{4} - 52 p T^{5} - 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5226 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 T - 100 T^{2} + 4 T^{3} + 7795 T^{4} + 4 p T^{5} - 100 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 252 T^{2} + 2592 T^{3} + 23627 T^{4} + 2592 p T^{5} + 252 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T + 16 T^{2} - 24 T^{3} - 2973 T^{4} - 24 p T^{5} + 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 7 T^{2} - 108 T^{3} + 7536 T^{4} - 108 p T^{5} - 7 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 22 T + 240 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 27891 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 91 T^{2} + 234 T^{3} + 6252 T^{4} + 234 p T^{5} - 91 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 14970 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.10291904435687539948480390299, −6.69896982824569866357980805914, −6.59552415026321685950301952584, −6.37613647204296684491145733113, −6.33766179214080872604898434387, −6.23172776672505485599274758870, −5.80482597388852968616672723722, −5.22958238431010645835022586089, −5.18074505051056525125209611008, −5.00408859213917849017236354246, −4.81981029267178960552202250848, −4.25043746480939575025856416942, −4.23390965597585915922729339663, −4.00834013000594444079002460367, −3.75260233982136034757390941710, −3.48298653711297627676265603850, −3.41589326525930644051298024250, −2.76449855482346181568430756024, −2.54402970403787914260611956347, −2.23208521291072750141981471662, −2.06865745465827218817956831358, −1.97599574757692054837633050136, −1.30914055547878388529292974150, −1.10702026584507722750464599558, −0.50331140092112823942457754294,
0.50331140092112823942457754294, 1.10702026584507722750464599558, 1.30914055547878388529292974150, 1.97599574757692054837633050136, 2.06865745465827218817956831358, 2.23208521291072750141981471662, 2.54402970403787914260611956347, 2.76449855482346181568430756024, 3.41589326525930644051298024250, 3.48298653711297627676265603850, 3.75260233982136034757390941710, 4.00834013000594444079002460367, 4.23390965597585915922729339663, 4.25043746480939575025856416942, 4.81981029267178960552202250848, 5.00408859213917849017236354246, 5.18074505051056525125209611008, 5.22958238431010645835022586089, 5.80482597388852968616672723722, 6.23172776672505485599274758870, 6.33766179214080872604898434387, 6.37613647204296684491145733113, 6.59552415026321685950301952584, 6.69896982824569866357980805914, 7.10291904435687539948480390299
