| L(s) = 1 | + 2·7-s + 6·9-s − 4·17-s − 16·23-s + 2·25-s + 16·31-s − 12·41-s + 16·47-s + 3·49-s + 12·63-s − 16·71-s + 12·73-s − 16·79-s + 27·81-s + 12·89-s + 28·97-s − 12·113-s − 8·119-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2·9-s − 0.970·17-s − 3.33·23-s + 2/5·25-s + 2.87·31-s − 1.87·41-s + 2.33·47-s + 3/7·49-s + 1.51·63-s − 1.89·71-s + 1.40·73-s − 1.80·79-s + 3·81-s + 1.27·89-s + 2.84·97-s − 1.12·113-s − 0.733·119-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.510737926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510737926\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.26001053216394522954091653823, −10.02802462690031950476795924093, −9.738954303920613400659949004480, −8.943860890634936168755032201344, −8.662483801228599624285453916280, −8.198751758066426565343521706796, −7.68733378258482915238904640559, −7.53356222130215830201456907092, −6.90943940678629335274499558614, −6.35318431597181664960310599789, −6.25879022327998988420816635302, −5.51390009111885104010347936693, −4.79900662613161598031710452961, −4.49654318322456261879393212640, −4.14438652358631518155955229946, −3.75681814440807246550724075687, −2.78266886524975051263788886418, −2.01743277983972991638362598809, −1.75808336217274324287862295603, −0.790629161122765033494096088883,
0.790629161122765033494096088883, 1.75808336217274324287862295603, 2.01743277983972991638362598809, 2.78266886524975051263788886418, 3.75681814440807246550724075687, 4.14438652358631518155955229946, 4.49654318322456261879393212640, 4.79900662613161598031710452961, 5.51390009111885104010347936693, 6.25879022327998988420816635302, 6.35318431597181664960310599789, 6.90943940678629335274499558614, 7.53356222130215830201456907092, 7.68733378258482915238904640559, 8.198751758066426565343521706796, 8.662483801228599624285453916280, 8.943860890634936168755032201344, 9.738954303920613400659949004480, 10.02802462690031950476795924093, 10.26001053216394522954091653823
