| L(s) = 1 | − 5.31·2-s − 3·3-s + 20.2·4-s − 16.8·5-s + 15.9·6-s + 32.3·7-s − 64.7·8-s + 9·9-s + 89.5·10-s + 11·11-s − 60.6·12-s + 39.9·13-s − 171.·14-s + 50.5·15-s + 182.·16-s − 65.1·17-s − 47.7·18-s + 64.2·19-s − 340.·20-s − 97.0·21-s − 58.4·22-s + 138.·23-s + 194.·24-s + 159.·25-s − 211.·26-s − 27·27-s + 653.·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.52·4-s − 1.50·5-s + 1.08·6-s + 1.74·7-s − 2.86·8-s + 0.333·9-s + 2.83·10-s + 0.301·11-s − 1.45·12-s + 0.851·13-s − 3.28·14-s + 0.870·15-s + 2.85·16-s − 0.929·17-s − 0.625·18-s + 0.776·19-s − 3.80·20-s − 1.00·21-s − 0.566·22-s + 1.25·23-s + 1.65·24-s + 1.27·25-s − 1.59·26-s − 0.192·27-s + 4.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8285467170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8285467170\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 5.31T + 8T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 13 | \( 1 - 39.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 67.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 66.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 365.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 17.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 633.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 622.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 333.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 717.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 766.T + 9.12e5T^{2} \) |
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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
−8.620187817844654626325364673615, −8.195375283712939055528247425194, −7.32942860736090615587355279356, −7.09202302958011834534323230749, −5.85726283028225799263590947141, −4.74706484638215884730504738538, −3.81759701432448201832619879445, −2.41514089782523399605849390387, −1.17779905760531736883126753965, −0.71305424626469697007485172193,
0.71305424626469697007485172193, 1.17779905760531736883126753965, 2.41514089782523399605849390387, 3.81759701432448201832619879445, 4.74706484638215884730504738538, 5.85726283028225799263590947141, 7.09202302958011834534323230749, 7.32942860736090615587355279356, 8.195375283712939055528247425194, 8.620187817844654626325364673615
