| L(s) = 1 | + 0.801·2-s + 3·3-s − 7.35·4-s + 19.8·5-s + 2.40·6-s + 26.4·7-s − 12.3·8-s + 9·9-s + 15.9·10-s + 11·11-s − 22.0·12-s + 36.4·13-s + 21.1·14-s + 59.5·15-s + 48.9·16-s − 19.2·17-s + 7.21·18-s + 44.2·19-s − 146.·20-s + 79.2·21-s + 8.81·22-s − 41.0·23-s − 36.9·24-s + 269.·25-s + 29.2·26-s + 27·27-s − 194.·28-s + ⋯ |
| L(s) = 1 | + 0.283·2-s + 0.577·3-s − 0.919·4-s + 1.77·5-s + 0.163·6-s + 1.42·7-s − 0.543·8-s + 0.333·9-s + 0.503·10-s + 0.301·11-s − 0.530·12-s + 0.778·13-s + 0.404·14-s + 1.02·15-s + 0.765·16-s − 0.274·17-s + 0.0944·18-s + 0.534·19-s − 1.63·20-s + 0.823·21-s + 0.0854·22-s − 0.372·23-s − 0.314·24-s + 2.15·25-s + 0.220·26-s + 0.192·27-s − 1.31·28-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
| \(L(2)\) |
\(\approx\) |
\(5.109494058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.109494058\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.801T + 8T^{2} \) |
| 5 | \( 1 - 19.8T + 125T^{2} \) |
| 7 | \( 1 - 26.4T + 343T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 50.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 34.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 482.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 113.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 182.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 442.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 690.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 990.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 468.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 772.T + 9.12e5T^{2} \) |
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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
−8.869569944732231797069574381039, −8.308957530938221984261116997954, −7.30153553085993948253216171379, −6.13209432945942236847681542083, −5.54156952028456875755273883865, −4.80090594445104231496391262257, −3.99598896867131992359878981707, −2.80086014078782314991884314467, −1.76848833154333486467536705951, −1.10282460050394417985823620033,
1.10282460050394417985823620033, 1.76848833154333486467536705951, 2.80086014078782314991884314467, 3.99598896867131992359878981707, 4.80090594445104231496391262257, 5.54156952028456875755273883865, 6.13209432945942236847681542083, 7.30153553085993948253216171379, 8.308957530938221984261116997954, 8.869569944732231797069574381039
