| L(s) = 1 | + 1.39·2-s − 3·3-s − 6.05·4-s − 16.3·5-s − 4.18·6-s − 3.89·7-s − 19.6·8-s + 9·9-s − 22.7·10-s + 11·11-s + 18.1·12-s + 3.52·13-s − 5.43·14-s + 48.9·15-s + 21.0·16-s − 98.4·17-s + 12.5·18-s − 107.·19-s + 98.7·20-s + 11.6·21-s + 15.3·22-s + 92.6·23-s + 58.8·24-s + 141.·25-s + 4.92·26-s − 27·27-s + 23.5·28-s + ⋯ |
| L(s) = 1 | + 0.493·2-s − 0.577·3-s − 0.756·4-s − 1.45·5-s − 0.284·6-s − 0.210·7-s − 0.866·8-s + 0.333·9-s − 0.720·10-s + 0.301·11-s + 0.436·12-s + 0.0752·13-s − 0.103·14-s + 0.842·15-s + 0.328·16-s − 1.40·17-s + 0.164·18-s − 1.30·19-s + 1.10·20-s + 0.121·21-s + 0.148·22-s + 0.840·23-s + 0.500·24-s + 1.13·25-s + 0.0371·26-s − 0.192·27-s + 0.158·28-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 | \( 1 - 1.39T + 8T^{2} \) |
| 5 | \( 1 + 16.3T + 125T^{2} \) |
| 7 | \( 1 + 3.89T + 343T^{2} \) |
| 13 | \( 1 - 3.52T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 92.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 94.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 38.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 185.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 329.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 108.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 798.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 230.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 16.1T + 7.04e5T^{2} \) |
| 97 | \( 1 - 382.T + 9.12e5T^{2} \) |
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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
−8.515970438872191921762630228343, −7.62098657447097712499104579551, −6.64153177405885649736673276059, −6.09744491907134762927678296081, −4.76045322067031799211699662070, −4.44112530403322476076870746512, −3.74096725766384492338749719825, −2.65393121518901569287202690467, −0.824114722399317141717304842427, 0,
0.824114722399317141717304842427, 2.65393121518901569287202690467, 3.74096725766384492338749719825, 4.44112530403322476076870746512, 4.76045322067031799211699662070, 6.09744491907134762927678296081, 6.64153177405885649736673276059, 7.62098657447097712499104579551, 8.515970438872191921762630228343
