| L(s) = 1 | + 8·3-s + 46·9-s − 4·11-s − 16·16-s + 48·17-s − 48·19-s + 184·27-s − 32·33-s + 96·41-s + 28·43-s − 128·48-s + 384·51-s − 384·57-s + 164·59-s − 336·67-s − 48·73-s + 610·81-s + 316·83-s − 52·97-s − 184·99-s − 188·107-s + 288·113-s − 38·121-s + 768·123-s + 127-s + 224·129-s + 131-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 46/9·9-s − 0.363·11-s − 16-s + 2.82·17-s − 2.52·19-s + 6.81·27-s − 0.969·33-s + 2.34·41-s + 0.651·43-s − 8/3·48-s + 7.52·51-s − 6.73·57-s + 2.77·59-s − 5.01·67-s − 0.657·73-s + 7.53·81-s + 3.80·83-s − 0.536·97-s − 1.85·99-s − 1.75·107-s + 2.54·113-s − 0.314·121-s + 6.24·123-s + 0.00787·127-s + 1.73·129-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.749886630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.749886630\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )( 1 - 14 T^{2} + p^{4} T^{4} ) \) |
| 5 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2}( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 96 T + 4608 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )( 1 - 1246 T^{2} + p^{4} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2}( 1 - 3502 T^{2} + p^{4} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2}( 1 - 238 T^{2} + p^{4} T^{4} ) \) |
| 61 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 168 T + 14112 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 9506 T^{2} + p^{4} T^{4} )( 1 + 48 T + 1152 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 79 | $C_4\times C_2$ | \( 1 + p^{8} T^{8} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 - 158 T + p^{2} T^{2} )^{2}( 1 + 11186 T^{2} + p^{4} T^{4} ) \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 144 T + 10368 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )( 1 + 144 T + 10368 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 + 240 T + 28800 T^{2} + 240 p^{2} T^{3} + p^{4} T^{4} ) \) |
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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
−9.577818393296944042431351500232, −9.012293911230694027503275125947, −8.873923019086010266594309952647, −8.754025838858279168968362203563, −8.480130838900794845211974277751, −7.82827333059112851143619120401, −7.76079500438245568060487434296, −7.74595638277178403085855169706, −7.51970695530119147373301863062, −6.96602649718616442202683601754, −6.78753434747885517419431254722, −6.36830439734850330280549873593, −5.94459768317007640324851040212, −5.63606386168935680502083076707, −5.26224964581076400386554171282, −4.42070166315467217079051243674, −4.35677885857204447899535878392, −4.33215190713590144985705674940, −3.63235864568933165201782589634, −3.40950429531687631888185269321, −2.95081434397453112244995333998, −2.48271696374079102226764754692, −2.13947562892073897864002644124, −1.72417138947243909755814456860, −0.943491278847577971844287067744,
0.943491278847577971844287067744, 1.72417138947243909755814456860, 2.13947562892073897864002644124, 2.48271696374079102226764754692, 2.95081434397453112244995333998, 3.40950429531687631888185269321, 3.63235864568933165201782589634, 4.33215190713590144985705674940, 4.35677885857204447899535878392, 4.42070166315467217079051243674, 5.26224964581076400386554171282, 5.63606386168935680502083076707, 5.94459768317007640324851040212, 6.36830439734850330280549873593, 6.78753434747885517419431254722, 6.96602649718616442202683601754, 7.51970695530119147373301863062, 7.74595638277178403085855169706, 7.76079500438245568060487434296, 7.82827333059112851143619120401, 8.480130838900794845211974277751, 8.754025838858279168968362203563, 8.873923019086010266594309952647, 9.012293911230694027503275125947, 9.577818393296944042431351500232
