| L(s) = 1 | + 0.311·2-s + 2.90·3-s − 1.90·4-s + 5-s + 0.903·6-s − 4.42·7-s − 1.21·8-s + 5.42·9-s + 0.311·10-s − 2.62·11-s − 5.52·12-s + 0.474·13-s − 1.37·14-s + 2.90·15-s + 3.42·16-s + 5.05·17-s + 1.68·18-s − 19-s − 1.90·20-s − 12.8·21-s − 0.815·22-s − 1.37·23-s − 3.52·24-s + 25-s + 0.147·26-s + 7.05·27-s + 8.42·28-s + ⋯ |
| L(s) = 1 | + 0.219·2-s + 1.67·3-s − 0.951·4-s + 0.447·5-s + 0.368·6-s − 1.67·7-s − 0.429·8-s + 1.80·9-s + 0.0983·10-s − 0.790·11-s − 1.59·12-s + 0.131·13-s − 0.368·14-s + 0.749·15-s + 0.857·16-s + 1.22·17-s + 0.398·18-s − 0.229·19-s − 0.425·20-s − 2.80·21-s − 0.173·22-s − 0.287·23-s − 0.719·24-s + 0.200·25-s + 0.0289·26-s + 1.35·27-s + 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.352602430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.352602430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 - 0.474T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 4.42T + 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.88165428151931298038439437036, −13.10626848372171412707928666926, −12.66335678228115181574892050770, −10.07349265433879371920600464484, −9.604367651482477757989319265045, −8.670225780444789576473044835963, −7.49974145843644389566270094644, −5.77728124717938862488730915927, −3.84634828564337483485385365654, −2.86603342930793270402568953419,
2.86603342930793270402568953419, 3.84634828564337483485385365654, 5.77728124717938862488730915927, 7.49974145843644389566270094644, 8.670225780444789576473044835963, 9.604367651482477757989319265045, 10.07349265433879371920600464484, 12.66335678228115181574892050770, 13.10626848372171412707928666926, 13.88165428151931298038439437036
