| L(s) = 1 | − 4·2-s − 20.5·3-s + 16·4-s − 32.9·5-s + 82.1·6-s − 257.·7-s − 64·8-s + 179.·9-s + 131.·10-s − 56.7·11-s − 328.·12-s − 978.·13-s + 1.02e3·14-s + 677.·15-s + 256·16-s − 136.·17-s − 716.·18-s + 2.50e3·19-s − 527.·20-s + 5.28e3·21-s + 226.·22-s + 2.96e3·23-s + 1.31e3·24-s − 2.03e3·25-s + 3.91e3·26-s + 1.31e3·27-s − 4.11e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.31·3-s + 0.5·4-s − 0.589·5-s + 0.931·6-s − 1.98·7-s − 0.353·8-s + 0.736·9-s + 0.417·10-s − 0.141·11-s − 0.658·12-s − 1.60·13-s + 1.40·14-s + 0.777·15-s + 0.250·16-s − 0.114·17-s − 0.521·18-s + 1.59·19-s − 0.294·20-s + 2.61·21-s + 0.0999·22-s + 1.16·23-s + 0.465·24-s − 0.652·25-s + 1.13·26-s + 0.346·27-s − 0.992·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1926456845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1926456845\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 37 | \( 1 - 1.36e3T \) |
| good | 3 | \( 1 + 20.5T + 243T^{2} \) |
| 5 | \( 1 + 32.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 257.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 56.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 978.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 136.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.21e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 7.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.21e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.25e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 67.8T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.68e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.77e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.17e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.91e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.28e5T + 8.58e9T^{2} \) |
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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
−13.15956352353885907410715620786, −12.17556723249562308570932505037, −11.45632859451958482503990637797, −10.07951646629911104116968254328, −9.450974342746563655770232191365, −7.44085376384930019728468668317, −6.58608008990804492182526616403, −5.27734817810250443637074197233, −3.15625185671689225548352280809, −0.37497905598864052628362443396,
0.37497905598864052628362443396, 3.15625185671689225548352280809, 5.27734817810250443637074197233, 6.58608008990804492182526616403, 7.44085376384930019728468668317, 9.450974342746563655770232191365, 10.07951646629911104116968254328, 11.45632859451958482503990637797, 12.17556723249562308570932505037, 13.15956352353885907410715620786
