| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·5-s − 2·6-s − 2·7-s + 4·8-s − 4·9-s − 4·10-s − 3·12-s + 6·13-s − 4·14-s + 2·15-s + 5·16-s + 17-s − 8·18-s + 3·19-s − 6·20-s + 2·21-s − 8·23-s − 4·24-s + 3·25-s + 12·26-s + 6·27-s − 6·28-s − 8·29-s + 4·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s − 0.755·7-s + 1.41·8-s − 4/3·9-s − 1.26·10-s − 0.866·12-s + 1.66·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s + 0.242·17-s − 1.88·18-s + 0.688·19-s − 1.34·20-s + 0.436·21-s − 1.66·23-s − 0.816·24-s + 3/5·25-s + 2.35·26-s + 1.15·27-s − 1.13·28-s − 1.48·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 165 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 209 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 45 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−7.53712659283738338634522538154, −7.18947093485189085952419740836, −6.55851093629237740701352195286, −6.55479876292262841770399356059, −6.08707878204852748671902137713, −5.96207261299810892494366380299, −5.54832036920893135520857945650, −5.19275511384046089273203549713, −4.92483490060742295558197980737, −4.46062034475535623420946684046, −3.84484390413238008822231001962, −3.69207901477114918765426925616, −3.36111319641238744756435358516, −3.30880437563486219574014474742, −2.42194252831615710031511039339, −2.37026983234579516477833290512, −1.44556354007208217400048165626, −1.13961180205928820895961359180, 0, 0,
1.13961180205928820895961359180, 1.44556354007208217400048165626, 2.37026983234579516477833290512, 2.42194252831615710031511039339, 3.30880437563486219574014474742, 3.36111319641238744756435358516, 3.69207901477114918765426925616, 3.84484390413238008822231001962, 4.46062034475535623420946684046, 4.92483490060742295558197980737, 5.19275511384046089273203549713, 5.54832036920893135520857945650, 5.96207261299810892494366380299, 6.08707878204852748671902137713, 6.55479876292262841770399356059, 6.55851093629237740701352195286, 7.18947093485189085952419740836, 7.53712659283738338634522538154
