| L(s) = 1 | − 2·2-s − 8.14·3-s + 4·4-s + 12.6·5-s + 16.2·6-s + 28.7·7-s − 8·8-s + 39.3·9-s − 25.3·10-s + 52.8·11-s − 32.5·12-s − 57.5·14-s − 103.·15-s + 16·16-s + 71.0·17-s − 78.7·18-s − 65.7·19-s + 50.6·20-s − 234.·21-s − 105.·22-s − 44.4·23-s + 65.1·24-s + 35.3·25-s − 100.·27-s + 115.·28-s − 78.4·29-s + 206.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.56·3-s + 0.5·4-s + 1.13·5-s + 1.10·6-s + 1.55·7-s − 0.353·8-s + 1.45·9-s − 0.800·10-s + 1.44·11-s − 0.784·12-s − 1.09·14-s − 1.77·15-s + 0.250·16-s + 1.01·17-s − 1.03·18-s − 0.794·19-s + 0.566·20-s − 2.43·21-s − 1.02·22-s − 0.402·23-s + 0.554·24-s + 0.283·25-s − 0.719·27-s + 0.776·28-s − 0.502·29-s + 1.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.300951608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300951608\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 8.14T + 27T^{2} \) |
| 5 | \( 1 - 12.6T + 125T^{2} \) |
| 7 | \( 1 - 28.7T + 343T^{2} \) |
| 11 | \( 1 - 52.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 65.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 78.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 120.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 400.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 116.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 781.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 52.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 805.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 401.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 323.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 794.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 444.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 79.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 3.89T + 9.12e5T^{2} \) |
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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
−11.03020130807953683191247792347, −10.36455951970622868470413873965, −9.449383065648710418690852591542, −8.365904625949980666499908274045, −7.12068216186550186770945567698, −6.12330025802612361833090414777, −5.50480634987000672953430321126, −4.34243423991075556427195790885, −1.85151054322463600776368006137, −1.03028320616562102177409529056,
1.03028320616562102177409529056, 1.85151054322463600776368006137, 4.34243423991075556427195790885, 5.50480634987000672953430321126, 6.12330025802612361833090414777, 7.12068216186550186770945567698, 8.365904625949980666499908274045, 9.449383065648710418690852591542, 10.36455951970622868470413873965, 11.03020130807953683191247792347
