L(s) = 1 | − 5.27·2-s + 4.83·3-s + 19.8·4-s − 5·5-s − 25.4·6-s − 13.3·7-s − 62.3·8-s − 3.66·9-s + 26.3·10-s + 11·11-s + 95.7·12-s + 44.5·13-s + 70.2·14-s − 24.1·15-s + 170.·16-s + 128.·17-s + 19.3·18-s + 19·19-s − 99.0·20-s − 64.3·21-s − 58.0·22-s + 72.9·23-s − 301.·24-s + 25·25-s − 235.·26-s − 148.·27-s − 263.·28-s + ⋯ |
L(s) = 1 | − 1.86·2-s + 0.929·3-s + 2.47·4-s − 0.447·5-s − 1.73·6-s − 0.719·7-s − 2.75·8-s − 0.135·9-s + 0.833·10-s + 0.301·11-s + 2.30·12-s + 0.950·13-s + 1.34·14-s − 0.415·15-s + 2.65·16-s + 1.83·17-s + 0.252·18-s + 0.229·19-s − 1.10·20-s − 0.668·21-s − 0.562·22-s + 0.661·23-s − 2.56·24-s + 0.200·25-s − 1.77·26-s − 1.05·27-s − 1.78·28-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
\(L(2)\) |
\(\approx\) |
\(0.9810575856\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9810575856\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.27T + 8T^{2} \) |
| 3 | \( 1 - 4.83T + 27T^{2} \) |
| 7 | \( 1 + 13.3T + 343T^{2} \) |
| 13 | \( 1 - 44.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 128.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 72.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 197.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 540.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 334.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 910.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 854.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 277.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 534.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 787.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−9.447993931905894334095962190480, −8.746807195842748751396637189784, −8.047267077206532196295934311238, −7.51565230687462457745091651129, −6.57740355828185198555524861967, −5.65070339801017925454648885292, −3.51477499139076041131074112200, −3.10015163713564267290122166512, −1.75002750745644637584349107758, −0.66797565956129901812378318085,
0.66797565956129901812378318085, 1.75002750745644637584349107758, 3.10015163713564267290122166512, 3.51477499139076041131074112200, 5.65070339801017925454648885292, 6.57740355828185198555524861967, 7.51565230687462457745091651129, 8.047267077206532196295934311238, 8.746807195842748751396637189784, 9.447993931905894334095962190480