| L(s) = 1 | + 4·3-s − 4-s + 2·5-s + 6·7-s + 6·9-s − 6·11-s − 4·12-s + 8·15-s − 3·16-s + 4·19-s − 2·20-s + 24·21-s − 4·23-s + 3·25-s − 4·27-s − 6·28-s − 8·29-s + 14·31-s − 24·33-s + 12·35-s − 6·36-s + 16·37-s − 4·41-s + 6·43-s + 6·44-s + 12·45-s − 10·47-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1/2·4-s + 0.894·5-s + 2.26·7-s + 2·9-s − 1.80·11-s − 1.15·12-s + 2.06·15-s − 3/4·16-s + 0.917·19-s − 0.447·20-s + 5.23·21-s − 0.834·23-s + 3/5·25-s − 0.769·27-s − 1.13·28-s − 1.48·29-s + 2.51·31-s − 4.17·33-s + 2.02·35-s − 36-s + 2.63·37-s − 0.624·41-s + 0.914·43-s + 0.904·44-s + 1.78·45-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.995095421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.995095421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14 T + 108 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.50042946193024514044995317707, −11.20615978353297622973517611335, −10.69771891429095141871934952574, −9.996869528830340317298617969780, −9.630632127749904485036007895488, −9.372397616331073856200636406603, −8.618619732226194278026710708803, −8.457491591467318534347785994361, −7.942704257016916182597403110190, −7.77257920103062466103532041275, −7.47166417084046517551393735352, −6.36240588451021929172490602496, −5.50434754839128507584654245670, −5.35669868641989726360220757988, −4.39924457628057628906357696540, −4.36898714158056822267253444492, −3.15221834759892545408462460316, −2.57191551949308406678639843077, −2.29542586769049351946080678121, −1.52265949476872273257625846986,
1.52265949476872273257625846986, 2.29542586769049351946080678121, 2.57191551949308406678639843077, 3.15221834759892545408462460316, 4.36898714158056822267253444492, 4.39924457628057628906357696540, 5.35669868641989726360220757988, 5.50434754839128507584654245670, 6.36240588451021929172490602496, 7.47166417084046517551393735352, 7.77257920103062466103532041275, 7.942704257016916182597403110190, 8.457491591467318534347785994361, 8.618619732226194278026710708803, 9.372397616331073856200636406603, 9.630632127749904485036007895488, 9.996869528830340317298617969780, 10.69771891429095141871934952574, 11.20615978353297622973517611335, 11.50042946193024514044995317707
