| L(s) = 1 | + 8·4-s − 12·13-s + 23·16-s + 24·19-s + 68·25-s + 64·31-s − 92·37-s − 28·43-s + 14·49-s − 96·52-s + 96·61-s − 16·64-s + 156·67-s + 196·73-s + 192·76-s − 484·79-s + 16·97-s + 544·100-s + 80·103-s − 32·109-s − 52·121-s + 512·124-s + 127-s + 131-s + 137-s + 139-s − 736·148-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.923·13-s + 1.43·16-s + 1.26·19-s + 2.71·25-s + 2.06·31-s − 2.48·37-s − 0.651·43-s + 2/7·49-s − 1.84·52-s + 1.57·61-s − 1/4·64-s + 2.32·67-s + 2.68·73-s + 2.52·76-s − 6.12·79-s + 0.164·97-s + 5.43·100-s + 0.776·103-s − 0.293·109-s − 0.429·121-s + 4.12·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 4.97·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.144421982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.144421982\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{3} T^{2} + 41 T^{4} - p^{7} T^{6} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2231 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 26255 T^{4} + 52 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 6 T + 172 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 12 T + 751 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 964 T^{2} + 723399 T^{4} - 964 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 560 T^{2} + 1481762 T^{4} - 560 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 32 T + 1835 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 46 T + 1895 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 1983383 T^{4} - 212 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 2172 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7792 T^{2} + 24901890 T^{4} - 7792 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4612 T^{2} + 19057398 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 8251314 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 48 T - 74 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 78 T + 10492 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4972 T^{2} - 694929 T^{4} - 4972 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 98 T + 9356 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 242 T + 26556 T^{2} + 242 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2968 T^{2} + 90986226 T^{4} - 2968 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23084 T^{2} + 240217871 T^{4} - 23084 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 8726 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.808837469738253742789583324322, −8.599496540772907144937048503577, −8.441428314539462904843751559097, −8.161514052114040203208925141423, −7.935093765959198865242421203241, −7.19115274670839642728840404840, −7.18316611226507957954376393791, −7.03355254048878499851329517237, −7.00345307192654192995220586709, −6.55441226858031104003712705194, −6.35013629387965750045088782790, −5.89053352468769319973986974840, −5.57292957442234784375623711667, −5.16018856626205032548801493639, −5.10060500313364938215169746061, −4.52624379624049417877732965631, −4.46562978923733278612105752396, −3.63751491190862072636869344443, −3.35682206103020906336606590577, −3.02454824001006349512758653221, −2.52744408403579359690421031930, −2.51260793622614718424922784784, −1.83344230492164416300197464281, −1.32055352489388941270409839888, −0.69276970627616638311289019733,
0.69276970627616638311289019733, 1.32055352489388941270409839888, 1.83344230492164416300197464281, 2.51260793622614718424922784784, 2.52744408403579359690421031930, 3.02454824001006349512758653221, 3.35682206103020906336606590577, 3.63751491190862072636869344443, 4.46562978923733278612105752396, 4.52624379624049417877732965631, 5.10060500313364938215169746061, 5.16018856626205032548801493639, 5.57292957442234784375623711667, 5.89053352468769319973986974840, 6.35013629387965750045088782790, 6.55441226858031104003712705194, 7.00345307192654192995220586709, 7.03355254048878499851329517237, 7.18316611226507957954376393791, 7.19115274670839642728840404840, 7.935093765959198865242421203241, 8.161514052114040203208925141423, 8.441428314539462904843751559097, 8.599496540772907144937048503577, 8.808837469738253742789583324322
