| L(s) = 1 | − 0.236·2-s − 7.94·4-s + 15.2·5-s − 20.6·7-s + 3.76·8-s − 3.59·10-s + 7.63·11-s + 55.0·13-s + 4.87·14-s + 62.6·16-s + 72.7·17-s − 96.7·19-s − 121.·20-s − 1.80·22-s + 23·23-s + 107.·25-s − 13.0·26-s + 164.·28-s + 228.·29-s + 336.·31-s − 44.9·32-s − 17.1·34-s − 314.·35-s + 213.·37-s + 22.8·38-s + 57.3·40-s + 325.·41-s + ⋯ |
| L(s) = 1 | − 0.0834·2-s − 0.993·4-s + 1.36·5-s − 1.11·7-s + 0.166·8-s − 0.113·10-s + 0.209·11-s + 1.17·13-s + 0.0930·14-s + 0.979·16-s + 1.03·17-s − 1.16·19-s − 1.35·20-s − 0.0174·22-s + 0.208·23-s + 0.857·25-s − 0.0980·26-s + 1.10·28-s + 1.46·29-s + 1.94·31-s − 0.248·32-s − 0.0866·34-s − 1.51·35-s + 0.946·37-s + 0.0975·38-s + 0.226·40-s + 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.602081856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.602081856\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 0.236T + 8T^{2} \) |
| 5 | \( 1 - 15.2T + 125T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 11 | \( 1 - 7.63T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 96.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 336.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 297.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 494.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 220.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 502.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 69.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 140.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 273.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 234.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 880.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 20.4T + 9.12e5T^{2} \) |
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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
−12.26473734384424517465870120536, −10.53664978995765250686484920247, −9.907674480678983152732411617577, −9.147321816479382648611307364740, −8.227249603961742161210678483528, −6.40052265229256585779294249322, −5.88113510193971074688276552029, −4.40313437752440134788302497308, −2.97724174210892942791621363881, −1.04536357832079579036922131738,
1.04536357832079579036922131738, 2.97724174210892942791621363881, 4.40313437752440134788302497308, 5.88113510193971074688276552029, 6.40052265229256585779294249322, 8.227249603961742161210678483528, 9.147321816479382648611307364740, 9.907674480678983152732411617577, 10.53664978995765250686484920247, 12.26473734384424517465870120536
