| L(s) = 1 | + 6·3-s − 6·5-s − 10·7-s + 27·9-s + 16·11-s − 26·13-s − 36·15-s + 4·17-s + 70·19-s − 60·21-s − 128·23-s − 110·25-s + 108·27-s + 80·29-s − 250·31-s + 96·33-s + 60·35-s + 152·37-s − 156·39-s − 146·41-s + 504·43-s − 162·45-s − 524·47-s − 498·49-s + 24·51-s + 52·53-s − 96·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.536·5-s − 0.539·7-s + 9-s + 0.438·11-s − 0.554·13-s − 0.619·15-s + 0.0570·17-s + 0.845·19-s − 0.623·21-s − 1.16·23-s − 0.879·25-s + 0.769·27-s + 0.512·29-s − 1.44·31-s + 0.506·33-s + 0.289·35-s + 0.675·37-s − 0.640·39-s − 0.556·41-s + 1.78·43-s − 0.536·45-s − 1.62·47-s − 1.45·49-s + 0.0658·51-s + 0.134·53-s − 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 598 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 16 T + 918 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 70 T + 12118 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 80 T + 46310 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 250 T + 49782 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 152 T + 70470 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 146 T + 137634 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 504 T + 215286 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 524 T + 272222 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 52 T + 291198 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 164 T + 406182 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 304 T + 237958 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 914 T + 804838 T^{2} - 914 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 455470 T^{2} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 456 T + 825950 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 824 T + 1009374 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 828 T + 1032470 T^{2} + 828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 826 T + 1571354 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 552 T + 219630 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.237531694687323877835094608507, −8.015477672970482303670989439905, −7.65297196239295879219717400119, −7.47341296325385385718878937316, −6.75033180918405066368917242480, −6.73299933138166591858171015248, −6.11863476347586999451652403537, −5.64047112512998964043447999523, −5.22388813953890825643913289344, −4.73143305510029877797670287036, −4.10994378848860783003291260456, −3.94628964754630066782371273748, −3.47843182284336002331919620494, −3.14572912476690580172044778577, −2.45874610749956357644549805760, −2.26462285786915916032528024540, −1.47804494756137105180180355013, −1.12350226157738192658699375204, 0, 0,
1.12350226157738192658699375204, 1.47804494756137105180180355013, 2.26462285786915916032528024540, 2.45874610749956357644549805760, 3.14572912476690580172044778577, 3.47843182284336002331919620494, 3.94628964754630066782371273748, 4.10994378848860783003291260456, 4.73143305510029877797670287036, 5.22388813953890825643913289344, 5.64047112512998964043447999523, 6.11863476347586999451652403537, 6.73299933138166591858171015248, 6.75033180918405066368917242480, 7.47341296325385385718878937316, 7.65297196239295879219717400119, 8.015477672970482303670989439905, 8.237531694687323877835094608507
