| L(s) = 1 | + 3·4-s + 2·5-s − 9-s + 5·16-s + 6·20-s − 25-s − 2·29-s + 8·31-s − 3·36-s + 20·41-s − 2·45-s + 10·49-s − 8·59-s + 4·61-s + 3·64-s − 16·71-s − 16·79-s + 10·80-s + 81-s − 20·89-s − 3·100-s + 4·101-s + 28·109-s − 6·116-s − 22·121-s + 24·124-s − 12·125-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.894·5-s − 1/3·9-s + 5/4·16-s + 1.34·20-s − 1/5·25-s − 0.371·29-s + 1.43·31-s − 1/2·36-s + 3.12·41-s − 0.298·45-s + 10/7·49-s − 1.04·59-s + 0.512·61-s + 3/8·64-s − 1.89·71-s − 1.80·79-s + 1.11·80-s + 1/9·81-s − 2.11·89-s − 0.299·100-s + 0.398·101-s + 2.68·109-s − 0.557·116-s − 2·121-s + 2.15·124-s − 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.924411129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.924411129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.46703146214489623876729206357, −11.00608691485231665295111499680, −10.30850117503531401580472386251, −10.26811213835576722195467255803, −9.699035994704016622521748578347, −9.002089539462079661753590258004, −8.858554209804972962442823422743, −8.015895656487389401265565457087, −7.50945056583642494634706499276, −7.34519022737667832592625913677, −6.49736812590098311652648057645, −6.29898920478888041882346927315, −5.68843211675051030193084740788, −5.54235187190553773505443772454, −4.48179666436169288321200462777, −4.03626721994331605120916190742, −2.90221470476548853027593523716, −2.70725007613195789371249212216, −2.01746434099217470805047588555, −1.18931696640093760457797633098,
1.18931696640093760457797633098, 2.01746434099217470805047588555, 2.70725007613195789371249212216, 2.90221470476548853027593523716, 4.03626721994331605120916190742, 4.48179666436169288321200462777, 5.54235187190553773505443772454, 5.68843211675051030193084740788, 6.29898920478888041882346927315, 6.49736812590098311652648057645, 7.34519022737667832592625913677, 7.50945056583642494634706499276, 8.015895656487389401265565457087, 8.858554209804972962442823422743, 9.002089539462079661753590258004, 9.699035994704016622521748578347, 10.26811213835576722195467255803, 10.30850117503531401580472386251, 11.00608691485231665295111499680, 11.46703146214489623876729206357
