| L(s) = 1 | + 4·2-s − 20.1·3-s + 16·4-s + 57.7·5-s − 80.7·6-s + 142.·7-s + 64·8-s + 164.·9-s + 231.·10-s + 643.·11-s − 323.·12-s + 115.·13-s + 571.·14-s − 1.16e3·15-s + 256·16-s − 1.74e3·17-s + 658.·18-s − 361·19-s + 924.·20-s − 2.88e3·21-s + 2.57e3·22-s + 3.70e3·23-s − 1.29e3·24-s + 210.·25-s + 460.·26-s + 1.58e3·27-s + 2.28e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 1.03·5-s − 0.915·6-s + 1.10·7-s + 0.353·8-s + 0.677·9-s + 0.730·10-s + 1.60·11-s − 0.647·12-s + 0.189·13-s + 0.779·14-s − 1.33·15-s + 0.250·16-s − 1.46·17-s + 0.479·18-s − 0.229·19-s + 0.516·20-s − 1.42·21-s + 1.13·22-s + 1.45·23-s − 0.457·24-s + 0.0673·25-s + 0.133·26-s + 0.417·27-s + 0.551·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.093248175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093248175\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 + 20.1T + 243T^{2} \) |
| 5 | \( 1 - 57.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 142.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 643.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 115.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.74e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.70e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.57e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.30e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e3T + 8.58e9T^{2} \) |
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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
−15.09193929313047118974888352597, −14.07330896534582739522114063421, −12.85798146640900660363745645900, −11.45464697421242031975809755460, −10.98844985585254984354553594836, −9.133110885022791900535233447171, −6.78898380911210053438962378376, −5.76556100528320293582489464848, −4.53730493954727352679455580482, −1.59770935207162374156197745643,
1.59770935207162374156197745643, 4.53730493954727352679455580482, 5.76556100528320293582489464848, 6.78898380911210053438962378376, 9.133110885022791900535233447171, 10.98844985585254984354553594836, 11.45464697421242031975809755460, 12.85798146640900660363745645900, 14.07330896534582739522114063421, 15.09193929313047118974888352597
