| L(s) = 1 | + 2·5-s + 6·7-s + 12·11-s + 13-s + 2·17-s − 8·19-s + 5·25-s + 2·29-s − 2·31-s + 12·35-s − 14·37-s + 43-s + 13·49-s + 4·53-s + 24·55-s + 8·59-s + 11·61-s + 2·65-s − 15·67-s + 6·71-s − 9·73-s + 72·77-s + 13·79-s + 28·83-s + 4·85-s − 12·89-s + 6·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.26·7-s + 3.61·11-s + 0.277·13-s + 0.485·17-s − 1.83·19-s + 25-s + 0.371·29-s − 0.359·31-s + 2.02·35-s − 2.30·37-s + 0.152·43-s + 13/7·49-s + 0.549·53-s + 3.23·55-s + 1.04·59-s + 1.40·61-s + 0.248·65-s − 1.83·67-s + 0.712·71-s − 1.05·73-s + 8.20·77-s + 1.46·79-s + 3.07·83-s + 0.433·85-s − 1.27·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.268696066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.268696066\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.670489023489708317396972883091, −9.225625745886720171598510732026, −8.900400645572672298106023856480, −8.753297411822802100666184338691, −8.178746637089807812465327563453, −8.094166374400988116431568023200, −7.04571880996187750290626289361, −6.99302870625718669953158674130, −6.49966895804823778222046623604, −6.22687912254768452147324238119, −5.56703276868370906837850703674, −5.24568313561631342081293445516, −4.50668079186704118135919480250, −4.45821548773101626196950906517, −3.65555452104849576191233106448, −3.63118667314635298068719082385, −2.32184761195797885211662949271, −1.92137721713990289539439987616, −1.33633340468819628997221321433, −1.21663709903951145327394620484,
1.21663709903951145327394620484, 1.33633340468819628997221321433, 1.92137721713990289539439987616, 2.32184761195797885211662949271, 3.63118667314635298068719082385, 3.65555452104849576191233106448, 4.45821548773101626196950906517, 4.50668079186704118135919480250, 5.24568313561631342081293445516, 5.56703276868370906837850703674, 6.22687912254768452147324238119, 6.49966895804823778222046623604, 6.99302870625718669953158674130, 7.04571880996187750290626289361, 8.094166374400988116431568023200, 8.178746637089807812465327563453, 8.753297411822802100666184338691, 8.900400645572672298106023856480, 9.225625745886720171598510732026, 9.670489023489708317396972883091
