| L(s) = 1 | − 8·7-s + 5·9-s + 2·17-s − 8·23-s − 8·31-s + 6·41-s + 34·49-s − 40·63-s + 32·71-s − 22·73-s − 8·79-s + 16·81-s + 26·89-s + 28·97-s + 32·103-s − 18·113-s − 16·119-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 10·153-s + 157-s + 64·161-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 5/3·9-s + 0.485·17-s − 1.66·23-s − 1.43·31-s + 0.937·41-s + 34/7·49-s − 5.03·63-s + 3.79·71-s − 2.57·73-s − 0.900·79-s + 16/9·81-s + 2.75·89-s + 2.84·97-s + 3.15·103-s − 1.69·113-s − 1.46·119-s + 1.18·121-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 + 0.808·153-s + 0.0798·157-s + 5.04·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230773999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230773999\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 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
−8.947382563957632531029738487936, −8.683900733176620801326906485306, −7.945177252268149683089372246518, −7.57359039998172996244018338559, −7.24034427732830543446347290684, −7.14008776231469783459411466559, −6.34442395602916811024833046607, −6.33508477062926971884000144381, −6.11653861590357841480132883650, −5.55745688727170862732983393474, −5.03447159456176851982786401304, −4.48291018168263801919329215056, −3.99262845648412737011119246262, −3.66733385458929269410796103707, −3.42210480882997463319333170467, −3.01593816962426168176258095917, −2.10845860047743782286436735804, −2.07661037991701221183873514962, −0.991862290738104516009066187911, −0.40026185881022263384421213635,
0.40026185881022263384421213635, 0.991862290738104516009066187911, 2.07661037991701221183873514962, 2.10845860047743782286436735804, 3.01593816962426168176258095917, 3.42210480882997463319333170467, 3.66733385458929269410796103707, 3.99262845648412737011119246262, 4.48291018168263801919329215056, 5.03447159456176851982786401304, 5.55745688727170862732983393474, 6.11653861590357841480132883650, 6.33508477062926971884000144381, 6.34442395602916811024833046607, 7.14008776231469783459411466559, 7.24034427732830543446347290684, 7.57359039998172996244018338559, 7.945177252268149683089372246518, 8.683900733176620801326906485306, 8.947382563957632531029738487936
