| L(s) = 1 | − 6·5-s + 4·7-s + 4·11-s − 5·13-s + 3·17-s + 4·19-s − 8·23-s + 17·25-s − 5·29-s − 16·31-s − 24·35-s − 7·37-s − 9·41-s − 8·43-s + 8·47-s + 7·49-s + 10·53-s − 24·55-s + 4·59-s + 5·61-s + 30·65-s − 8·67-s − 4·71-s + 22·73-s + 16·77-s − 8·79-s − 18·85-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1.51·7-s + 1.20·11-s − 1.38·13-s + 0.727·17-s + 0.917·19-s − 1.66·23-s + 17/5·25-s − 0.928·29-s − 2.87·31-s − 4.05·35-s − 1.15·37-s − 1.40·41-s − 1.21·43-s + 1.16·47-s + 49-s + 1.37·53-s − 3.23·55-s + 0.520·59-s + 0.640·61-s + 3.72·65-s − 0.977·67-s − 0.474·71-s + 2.57·73-s + 1.82·77-s − 0.900·79-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6928871698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6928871698\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.44576908600006344510224581554, −9.831099065397905593927316513367, −9.339559828881023943090777047827, −8.926709517664852580663735112682, −8.408689480637070077514118951523, −8.045785227277115288864906260437, −7.81178514871832325268213458267, −7.32847919310981351068343944314, −7.16333818514872748276182623727, −6.80406900819133450422673111009, −5.72284034130761734054914096761, −5.31308900094643306220052118701, −5.03438953992755654583183824456, −4.30107391680632332291633076927, −3.86837836122392383913578565062, −3.76094796415234917020729315861, −3.21063136376065098437512599043, −2.03891394506725681742493250815, −1.60646807934623313479652226127, −0.39906640497337272797831426704,
0.39906640497337272797831426704, 1.60646807934623313479652226127, 2.03891394506725681742493250815, 3.21063136376065098437512599043, 3.76094796415234917020729315861, 3.86837836122392383913578565062, 4.30107391680632332291633076927, 5.03438953992755654583183824456, 5.31308900094643306220052118701, 5.72284034130761734054914096761, 6.80406900819133450422673111009, 7.16333818514872748276182623727, 7.32847919310981351068343944314, 7.81178514871832325268213458267, 8.045785227277115288864906260437, 8.408689480637070077514118951523, 8.926709517664852580663735112682, 9.339559828881023943090777047827, 9.831099065397905593927316513367, 10.44576908600006344510224581554
