| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 19.0·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 38.1·10-s − 18.1·11-s + 12·12-s + 48.5·13-s − 14·14-s + 57.2·15-s + 16·16-s − 77.3·17-s + 18·18-s + 87.3·19-s + 76.3·20-s − 21·21-s − 36.2·22-s − 23·23-s + 24·24-s + 239.·25-s + 97.0·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.70·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.20·10-s − 0.496·11-s + 0.288·12-s + 1.03·13-s − 0.267·14-s + 0.985·15-s + 0.250·16-s − 1.10·17-s + 0.235·18-s + 1.05·19-s + 0.853·20-s − 0.218·21-s − 0.351·22-s − 0.208·23-s + 0.204·24-s + 1.91·25-s + 0.732·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.829868852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.829868852\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 19.0T + 125T^{2} \) |
| 11 | \( 1 + 18.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 14.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 208.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 687.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 59.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 565.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 130.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 59.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 49.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 214.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 721.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−9.584819854871414026479180468998, −9.014283242279139795878256049050, −7.950610549035662296823262172172, −6.78615617285074982544145772358, −6.12833570629530160122084535569, −5.39853647608269378468636288578, −4.32781361475865126415805426103, −3.07150829356546024015480916791, −2.33848106051866677812329791625, −1.27068469586609667188538888742,
1.27068469586609667188538888742, 2.33848106051866677812329791625, 3.07150829356546024015480916791, 4.32781361475865126415805426103, 5.39853647608269378468636288578, 6.12833570629530160122084535569, 6.78615617285074982544145772358, 7.950610549035662296823262172172, 9.014283242279139795878256049050, 9.584819854871414026479180468998
