| L(s) = 1 | + 2·2-s + 4·4-s + 13.0·5-s + 18.7·7-s + 8·8-s + 26.1·10-s + 41.1·11-s + 32.6·13-s + 37.4·14-s + 16·16-s − 76.8·17-s − 164.·19-s + 52.2·20-s + 82.2·22-s + 23·23-s + 45.5·25-s + 65.2·26-s + 74.8·28-s − 42.5·29-s + 246.·31-s + 32·32-s − 153.·34-s + 244.·35-s + 281.·37-s − 329.·38-s + 104.·40-s + 330.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.16·5-s + 1.01·7-s + 0.353·8-s + 0.825·10-s + 1.12·11-s + 0.696·13-s + 0.714·14-s + 0.250·16-s − 1.09·17-s − 1.99·19-s + 0.584·20-s + 0.796·22-s + 0.208·23-s + 0.364·25-s + 0.492·26-s + 0.505·28-s − 0.272·29-s + 1.42·31-s + 0.176·32-s − 0.775·34-s + 1.17·35-s + 1.25·37-s − 1.40·38-s + 0.412·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.276879431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.276879431\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 13.0T + 125T^{2} \) |
| 7 | \( 1 - 18.7T + 343T^{2} \) |
| 11 | \( 1 - 41.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 164.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 42.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 160.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 578.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 621.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 347.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 372.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 884.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 777.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 188.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 200.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 49.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 767.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.62e3T + 9.12e5T^{2} \) |
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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
−11.08986853437499792147636598825, −9.994029570591155972997483382277, −8.942865333554473318748254432574, −8.120969140279797155538721709164, −6.46602232825934006633131443330, −6.26191715678287388387211372077, −4.85448109818365890745536899559, −4.08056997116696408655359298016, −2.35921881247030632189357806540, −1.46409029141403172597851334610,
1.46409029141403172597851334610, 2.35921881247030632189357806540, 4.08056997116696408655359298016, 4.85448109818365890745536899559, 6.26191715678287388387211372077, 6.46602232825934006633131443330, 8.120969140279797155538721709164, 8.942865333554473318748254432574, 9.994029570591155972997483382277, 11.08986853437499792147636598825
