| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·5-s + 4·6-s + 4·8-s + 3·9-s + 8·10-s + 2·11-s + 6·12-s − 2·13-s + 8·15-s + 5·16-s + 2·17-s + 6·18-s + 10·19-s + 12·20-s + 4·22-s + 8·24-s + 4·25-s − 4·26-s + 4·27-s + 2·29-s + 16·30-s + 6·32-s + 4·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s + 1.63·6-s + 1.41·8-s + 9-s + 2.52·10-s + 0.603·11-s + 1.73·12-s − 0.554·13-s + 2.06·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 2.29·19-s + 2.68·20-s + 0.852·22-s + 1.63·24-s + 4/5·25-s − 0.784·26-s + 0.769·27-s + 0.371·29-s + 2.92·30-s + 1.06·32-s + 0.696·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.60856869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.60856869\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 93 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 129 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 127 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.629243093350739789727097178171, −8.374147696035896582432613341195, −7.64230264662072331506549624389, −7.59293193685483276554368958481, −7.11807322484073014623242411570, −6.85734238982899370682010520285, −6.25915914814804895108321497983, −6.08934666023661149400415239502, −5.55098167671041131397415648604, −5.35639526990505694452046802057, −4.93987988213078821040913320483, −4.59249532232073551331564012937, −3.90133318381047795293898974819, −3.68884533360026957461492748793, −3.05177542792737903772090459892, −2.94466706443188630048931573478, −2.30269776782336020683090928412, −2.02952332092683050589231995322, −1.38885018051565769559417429259, −1.09755613866173283200745994117,
1.09755613866173283200745994117, 1.38885018051565769559417429259, 2.02952332092683050589231995322, 2.30269776782336020683090928412, 2.94466706443188630048931573478, 3.05177542792737903772090459892, 3.68884533360026957461492748793, 3.90133318381047795293898974819, 4.59249532232073551331564012937, 4.93987988213078821040913320483, 5.35639526990505694452046802057, 5.55098167671041131397415648604, 6.08934666023661149400415239502, 6.25915914814804895108321497983, 6.85734238982899370682010520285, 7.11807322484073014623242411570, 7.59293193685483276554368958481, 7.64230264662072331506549624389, 8.374147696035896582432613341195, 8.629243093350739789727097178171
