L(s) = 1 | − 1.99·2-s + 2.62·3-s + 1.99·4-s + 5-s − 5.23·6-s + 2.62·7-s + 0.0113·8-s + 3.86·9-s − 1.99·10-s + 11-s + 5.22·12-s − 1.22·13-s − 5.24·14-s + 2.62·15-s − 4.01·16-s − 6.75·17-s − 7.72·18-s + 5.20·19-s + 1.99·20-s + 6.87·21-s − 1.99·22-s − 0.411·23-s + 0.0296·24-s + 25-s + 2.45·26-s + 2.27·27-s + 5.23·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.51·3-s + 0.997·4-s + 0.447·5-s − 2.13·6-s + 0.991·7-s + 0.00399·8-s + 1.28·9-s − 0.632·10-s + 0.301·11-s + 1.50·12-s − 0.340·13-s − 1.40·14-s + 0.676·15-s − 1.00·16-s − 1.63·17-s − 1.82·18-s + 1.19·19-s + 0.445·20-s + 1.50·21-s − 0.426·22-s − 0.0857·23-s + 0.00604·24-s + 0.200·25-s + 0.480·26-s + 0.437·27-s + 0.988·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.016027020\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.016027020\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 + 0.411T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 5.13T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 0.917T + 67T^{2} \) |
| 71 | \( 1 - 2.96T + 71T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−8.711457662393581622397611034600, −7.924529355059207491651054620794, −7.33683709997526673778752123203, −6.81604116882893966820791378056, −5.46126473530410941109732363062, −4.52919350228542382000482595980, −3.67656064365877400292946580269, −2.32215274802455845920280877370, −2.09852500237912757911947932054, −0.973996650221403173555680637689,
0.973996650221403173555680637689, 2.09852500237912757911947932054, 2.32215274802455845920280877370, 3.67656064365877400292946580269, 4.52919350228542382000482595980, 5.46126473530410941109732363062, 6.81604116882893966820791378056, 7.33683709997526673778752123203, 7.924529355059207491651054620794, 8.711457662393581622397611034600