| L(s) = 1 | + 2·4-s − 14·7-s − 4·13-s − 46·19-s − 32·25-s − 28·28-s − 94·31-s + 110·37-s + 92·43-s + 49·49-s − 8·52-s − 208·61-s − 8·64-s + 194·67-s − 130·73-s − 92·76-s − 226·79-s + 56·91-s + 416·97-s − 64·100-s − 238·103-s + 98·109-s − 80·121-s − 188·124-s + 127-s + 131-s + 644·133-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2·7-s − 0.307·13-s − 2.42·19-s − 1.27·25-s − 28-s − 3.03·31-s + 2.97·37-s + 2.13·43-s + 49-s − 0.153·52-s − 3.40·61-s − 1/8·64-s + 2.89·67-s − 1.78·73-s − 1.21·76-s − 2.86·79-s + 8/13·91-s + 4.28·97-s − 0.639·100-s − 2.31·103-s + 0.899·109-s − 0.661·121-s − 1.51·124-s + 0.00787·127-s + 0.00763·131-s + 4.84·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3286709268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3286709268\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 80 T^{2} - 8241 T^{4} + 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 290 T^{2} + 579 T^{4} + 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 23 T + 168 T^{2} + 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 47 T + 1248 T^{2} + 47 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 55 T + 1656 T^{2} - 55 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 4400 T^{2} + 14480319 T^{4} + 4400 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3026 T^{2} + 1266195 T^{4} + 3026 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 82 T + 3243 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} )( 1 + 82 T + 3243 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + 104 T + 7095 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 97 T + 4920 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 65 T - 1104 T^{2} + 65 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 113 T + 6528 T^{2} + 113 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12896 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
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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
−9.599756581471245348419445708532, −9.216004784049763995130623504540, −9.132350623213221004722475238012, −9.047645845221223444874556426567, −8.342429903846735060132612876899, −8.335795268186532813134327018418, −7.78448208418965535113255957010, −7.34352850029511816582050413624, −7.28160025789763728350140594101, −7.23526450570688817161070108714, −6.47197206452024649484117895631, −6.19178104287415572690315553211, −6.17068236854293091036916242674, −5.88063584730811461727270881512, −5.69095170645460252642782514380, −4.82183841278427246799468829340, −4.69563253193714826136105846030, −4.04337837454569100935481376714, −3.93557655546868593158299665112, −3.52607685136252793837716336924, −2.98883891574551189972239893149, −2.41770157712237887796399919123, −2.32752233641601650097612784257, −1.52330927143123835174096723099, −0.21824351510762451539807575722,
0.21824351510762451539807575722, 1.52330927143123835174096723099, 2.32752233641601650097612784257, 2.41770157712237887796399919123, 2.98883891574551189972239893149, 3.52607685136252793837716336924, 3.93557655546868593158299665112, 4.04337837454569100935481376714, 4.69563253193714826136105846030, 4.82183841278427246799468829340, 5.69095170645460252642782514380, 5.88063584730811461727270881512, 6.17068236854293091036916242674, 6.19178104287415572690315553211, 6.47197206452024649484117895631, 7.23526450570688817161070108714, 7.28160025789763728350140594101, 7.34352850029511816582050413624, 7.78448208418965535113255957010, 8.335795268186532813134327018418, 8.342429903846735060132612876899, 9.047645845221223444874556426567, 9.132350623213221004722475238012, 9.216004784049763995130623504540, 9.599756581471245348419445708532
