| L(s) = 1 | + 6·9-s − 12·11-s + 4·19-s + 10·25-s − 16·31-s − 24·41-s + 12·49-s + 24·59-s − 20·61-s − 24·71-s − 8·79-s + 14·81-s + 24·89-s − 72·99-s + 12·101-s − 8·109-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + ⋯ |
| L(s) = 1 | + 2·9-s − 3.61·11-s + 0.917·19-s + 2·25-s − 2.87·31-s − 3.74·41-s + 12/7·49-s + 3.12·59-s − 2.56·61-s − 2.84·71-s − 0.900·79-s + 14/9·81-s + 2.54·89-s − 7.23·99-s + 1.19·101-s − 0.766·109-s + 5.09·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.151222788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.151222788\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 22 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6 T^{2} + 302 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 662 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 4542 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 11454 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 142 T^{2} + 10374 T^{4} - 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5622 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 10662 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_4$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 174 T^{2} + 18782 T^{4} - 174 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.191259016259086839951194102281, −7.78090352218386426196057524651, −7.62948553413514924175436173275, −7.40744180709567331132945366982, −7.37079396287498764148208787598, −7.29162301662007195457792502083, −6.66240158823599218079173816162, −6.52667752527951512050078391828, −6.48724781096419677549843793191, −5.60233128917662150501563586196, −5.49154529873637549125296879108, −5.45324970390636644061466481841, −5.16370653955465194142215282014, −4.90856598343253198185912743718, −4.61594104495603038059802412130, −4.39665493458959059464216497439, −3.73970299622832026259108721631, −3.71737393637441797905767167907, −3.23214929709971508899963411252, −2.83598098964990282632676514655, −2.70591854902366443509550573097, −2.15682926332740490322782216036, −1.69017227416043172698776581242, −1.42390425852460577838482783963, −0.41877617072356171753293268452,
0.41877617072356171753293268452, 1.42390425852460577838482783963, 1.69017227416043172698776581242, 2.15682926332740490322782216036, 2.70591854902366443509550573097, 2.83598098964990282632676514655, 3.23214929709971508899963411252, 3.71737393637441797905767167907, 3.73970299622832026259108721631, 4.39665493458959059464216497439, 4.61594104495603038059802412130, 4.90856598343253198185912743718, 5.16370653955465194142215282014, 5.45324970390636644061466481841, 5.49154529873637549125296879108, 5.60233128917662150501563586196, 6.48724781096419677549843793191, 6.52667752527951512050078391828, 6.66240158823599218079173816162, 7.29162301662007195457792502083, 7.37079396287498764148208787598, 7.40744180709567331132945366982, 7.62948553413514924175436173275, 7.78090352218386426196057524651, 8.191259016259086839951194102281
