L(s) = 1 | − 2.70·2-s − 3·3-s − 0.664·4-s + 8.28·5-s + 8.12·6-s − 34.8·7-s + 23.4·8-s + 9·9-s − 22.4·10-s + 11·11-s + 1.99·12-s + 57.5·13-s + 94.3·14-s − 24.8·15-s − 58.2·16-s + 40.3·17-s − 24.3·18-s − 38.1·19-s − 5.50·20-s + 104.·21-s − 29.7·22-s − 42.0·23-s − 70.3·24-s − 56.3·25-s − 155.·26-s − 27·27-s + 23.1·28-s + ⋯ |
L(s) = 1 | − 0.957·2-s − 0.577·3-s − 0.0830·4-s + 0.741·5-s + 0.552·6-s − 1.88·7-s + 1.03·8-s + 0.333·9-s − 0.709·10-s + 0.301·11-s + 0.0479·12-s + 1.22·13-s + 1.80·14-s − 0.427·15-s − 0.910·16-s + 0.575·17-s − 0.319·18-s − 0.461·19-s − 0.0615·20-s + 1.08·21-s − 0.288·22-s − 0.381·23-s − 0.598·24-s − 0.450·25-s − 1.17·26-s − 0.192·27-s + 0.156·28-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 + 2.70T + 8T^{2} \) |
| 5 | \( 1 - 8.28T + 125T^{2} \) |
| 7 | \( 1 + 34.8T + 343T^{2} \) |
| 13 | \( 1 - 57.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 227.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 157.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 410.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 467.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.17e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 571.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.64e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 493.T + 9.12e5T^{2} \) |
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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
−8.680009434357326815363988073504, −7.65423701739392045135509340751, −6.77696254473031648980276342460, −6.07761417608748493644152467006, −5.58269852508051295093787213379, −4.14338681913727530032495110006, −3.45303756992748392040729207697, −2.02587096263595306386919098653, −0.934048999349563237573609029716, 0,
0.934048999349563237573609029716, 2.02587096263595306386919098653, 3.45303756992748392040729207697, 4.14338681913727530032495110006, 5.58269852508051295093787213379, 6.07761417608748493644152467006, 6.77696254473031648980276342460, 7.65423701739392045135509340751, 8.680009434357326815363988073504