L(s) = 1 | − 1.82·2-s + 5.45·3-s − 4.67·4-s + 5·5-s − 9.94·6-s + 19.7·7-s + 23.1·8-s + 2.78·9-s − 9.11·10-s + 11·11-s − 25.5·12-s − 35.9·13-s − 35.9·14-s + 27.2·15-s − 4.69·16-s − 72.8·17-s − 5.08·18-s + 19·19-s − 23.3·20-s + 107.·21-s − 20.0·22-s − 204.·23-s + 126.·24-s + 25·25-s + 65.4·26-s − 132.·27-s − 92.1·28-s + ⋯ |
L(s) = 1 | − 0.644·2-s + 1.05·3-s − 0.584·4-s + 0.447·5-s − 0.676·6-s + 1.06·7-s + 1.02·8-s + 0.103·9-s − 0.288·10-s + 0.301·11-s − 0.614·12-s − 0.766·13-s − 0.685·14-s + 0.469·15-s − 0.0733·16-s − 1.03·17-s − 0.0665·18-s + 0.229·19-s − 0.261·20-s + 1.11·21-s − 0.194·22-s − 1.85·23-s + 1.07·24-s + 0.200·25-s + 0.493·26-s − 0.941·27-s − 0.622·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 1.82T + 8T^{2} \) |
| 3 | \( 1 - 5.45T + 27T^{2} \) |
| 7 | \( 1 - 19.7T + 343T^{2} \) |
| 13 | \( 1 + 35.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 79.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 85.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 20.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 613.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 487.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 578.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 637.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 582.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 129.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 860.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 845.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 18.0T + 9.12e5T^{2} \) |
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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
−9.015834027336388884731518148863, −8.385863387846987965976708520469, −7.87971132357054710887243853186, −6.92613622278000801190416652794, −5.54093754959187883181279320568, −4.64587785071477958870069786917, −3.76850257284872217987627438100, −2.32120647854273339213290598690, −1.61636906542104148984014410611, 0,
1.61636906542104148984014410611, 2.32120647854273339213290598690, 3.76850257284872217987627438100, 4.64587785071477958870069786917, 5.54093754959187883181279320568, 6.92613622278000801190416652794, 7.87971132357054710887243853186, 8.385863387846987965976708520469, 9.015834027336388884731518148863