| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s − 7-s + 4·8-s + 3·9-s − 2·11-s + 6·12-s + 3·13-s − 2·14-s + 5·16-s + 5·17-s + 6·18-s + 10·19-s − 2·21-s − 4·22-s + 6·23-s + 8·24-s + 6·26-s + 4·27-s − 3·28-s + 29-s − 7·31-s + 6·32-s − 4·33-s + 10·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.377·7-s + 1.41·8-s + 9-s − 0.603·11-s + 1.73·12-s + 0.832·13-s − 0.534·14-s + 5/4·16-s + 1.21·17-s + 1.41·18-s + 2.29·19-s − 0.436·21-s − 0.852·22-s + 1.25·23-s + 1.63·24-s + 1.17·26-s + 0.769·27-s − 0.566·28-s + 0.185·29-s − 1.25·31-s + 1.06·32-s − 0.696·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.67746903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.67746903\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - T + 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_4$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 148 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 190 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.474927243537001538997605874057, −9.295076025066399803051015760500, −8.649900526990017520198593965492, −8.454550039795146646304185645619, −7.66021359414708904857439576447, −7.55782542928732899996886859552, −7.18947470221129138557556627783, −7.00197228143148131276307455673, −6.00183968544272493386025805274, −5.89262039950886530549282417099, −5.57953658586023276697550141185, −4.84146338921329697414164255965, −4.71093551869298797127752936069, −3.95915874394961630072595900300, −3.40856025364545543213370334391, −3.33988670955995520600324337706, −2.80078598940891654036773713627, −2.44959588397507033422295230038, −1.39521886447991673407175565192, −1.14866996728897251551600916234,
1.14866996728897251551600916234, 1.39521886447991673407175565192, 2.44959588397507033422295230038, 2.80078598940891654036773713627, 3.33988670955995520600324337706, 3.40856025364545543213370334391, 3.95915874394961630072595900300, 4.71093551869298797127752936069, 4.84146338921329697414164255965, 5.57953658586023276697550141185, 5.89262039950886530549282417099, 6.00183968544272493386025805274, 7.00197228143148131276307455673, 7.18947470221129138557556627783, 7.55782542928732899996886859552, 7.66021359414708904857439576447, 8.454550039795146646304185645619, 8.649900526990017520198593965492, 9.295076025066399803051015760500, 9.474927243537001538997605874057
