| L(s) = 1 | − 5.80·3-s + 4·4-s + 5·5-s + 10.2·7-s + 24.6·9-s − 20.1·11-s − 23.2·12-s − 29.0·15-s + 16·16-s + 1.04·17-s + 13.4·19-s + 20·20-s − 59.3·21-s + 25·25-s − 90.8·27-s + 40.9·28-s − 17·31-s + 116.·33-s + 51.1·35-s + 98.6·36-s + 67.5·37-s + 25.5·43-s − 80.4·44-s + 123.·45-s + 58.4·47-s − 92.8·48-s + 55.7·49-s + ⋯ |
| L(s) = 1 | − 1.93·3-s + 4-s + 5-s + 1.46·7-s + 2.73·9-s − 1.82·11-s − 1.93·12-s − 1.93·15-s + 16-s + 0.0615·17-s + 0.706·19-s + 20-s − 2.82·21-s + 25-s − 3.36·27-s + 1.46·28-s − 0.548·31-s + 3.53·33-s + 1.46·35-s + 2.73·36-s + 1.82·37-s + 0.594·43-s − 1.82·44-s + 2.73·45-s + 1.24·47-s − 1.93·48-s + 1.13·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.568202068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568202068\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 79 | \( 1 + 79T \) |
| good | 2 | \( 1 - 4T^{2} \) |
| 3 | \( 1 + 5.80T + 9T^{2} \) |
| 7 | \( 1 - 10.2T + 49T^{2} \) |
| 11 | \( 1 + 20.1T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 1.04T + 289T^{2} \) |
| 19 | \( 1 - 13.4T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 17T + 961T^{2} \) |
| 37 | \( 1 - 67.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 25.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 27.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 174.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.90038771604279121935899255392, −10.69225247873632440910105559752, −9.759558505411583861647452490542, −7.86336742206967313290425157129, −7.21512647352055497985338439262, −5.96695645956328703002978792751, −5.47640435396063387413609165025, −4.73200108274403313099754558032, −2.31692878639636401342297462016, −1.13986326295767458377217899329,
1.13986326295767458377217899329, 2.31692878639636401342297462016, 4.73200108274403313099754558032, 5.47640435396063387413609165025, 5.96695645956328703002978792751, 7.21512647352055497985338439262, 7.86336742206967313290425157129, 9.759558505411583861647452490542, 10.69225247873632440910105559752, 10.90038771604279121935899255392
