| L(s) = 1 | + 2·2-s + 3·3-s + 3·4-s + 6·6-s + 7-s + 4·8-s + 2·9-s + 2·11-s + 9·12-s − 2·13-s + 2·14-s + 5·16-s + 6·17-s + 4·18-s + 3·21-s + 4·22-s + 3·23-s + 12·24-s − 4·26-s − 6·27-s + 3·28-s + 5·29-s + 4·31-s + 6·32-s + 6·33-s + 12·34-s + 6·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 3/2·4-s + 2.44·6-s + 0.377·7-s + 1.41·8-s + 2/3·9-s + 0.603·11-s + 2.59·12-s − 0.554·13-s + 0.534·14-s + 5/4·16-s + 1.45·17-s + 0.942·18-s + 0.654·21-s + 0.852·22-s + 0.625·23-s + 2.44·24-s − 0.784·26-s − 1.15·27-s + 0.566·28-s + 0.928·29-s + 0.718·31-s + 1.06·32-s + 1.04·33-s + 2.05·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7562500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7562500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.24264100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.24264100\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 27 T + 347 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 233 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.824375699549168786323009890622, −8.575509505239067890168223835406, −8.190993133586674995743693269901, −7.86879946186261963453111309773, −7.38146257974561057783848267818, −7.37647521384077983434298589026, −6.66040837614641463550074002904, −6.28565867765117421281176164438, −5.79057350323019324878086767309, −5.61083243267716322283653389041, −4.88193816073991225703627574743, −4.72186208445271243408845837065, −4.09129671889804268104658275753, −3.85599478951911115318034105907, −3.24778173877597201328236647586, −2.96049512928031417486022759256, −2.58770742306666779220078870735, −2.28942256594814126408427138360, −1.46018589349766774281830868782, −0.972034215320911580751866605660,
0.972034215320911580751866605660, 1.46018589349766774281830868782, 2.28942256594814126408427138360, 2.58770742306666779220078870735, 2.96049512928031417486022759256, 3.24778173877597201328236647586, 3.85599478951911115318034105907, 4.09129671889804268104658275753, 4.72186208445271243408845837065, 4.88193816073991225703627574743, 5.61083243267716322283653389041, 5.79057350323019324878086767309, 6.28565867765117421281176164438, 6.66040837614641463550074002904, 7.37647521384077983434298589026, 7.38146257974561057783848267818, 7.86879946186261963453111309773, 8.190993133586674995743693269901, 8.575509505239067890168223835406, 8.824375699549168786323009890622
