| L(s) = 1 | + 4.11·2-s + 3·3-s + 8.90·4-s − 9.56·5-s + 12.3·6-s − 24.0·7-s + 3.72·8-s + 9·9-s − 39.3·10-s − 11·11-s + 26.7·12-s + 69.5·13-s − 98.7·14-s − 28.6·15-s − 55.9·16-s − 39.8·17-s + 37.0·18-s − 18.4·19-s − 85.1·20-s − 72.0·21-s − 45.2·22-s + 192.·23-s + 11.1·24-s − 33.5·25-s + 286.·26-s + 27·27-s − 213.·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.577·3-s + 1.11·4-s − 0.855·5-s + 0.839·6-s − 1.29·7-s + 0.164·8-s + 0.333·9-s − 1.24·10-s − 0.301·11-s + 0.642·12-s + 1.48·13-s − 1.88·14-s − 0.493·15-s − 0.873·16-s − 0.568·17-s + 0.484·18-s − 0.223·19-s − 0.952·20-s − 0.748·21-s − 0.438·22-s + 1.74·23-s + 0.0951·24-s − 0.268·25-s + 2.15·26-s + 0.192·27-s − 1.44·28-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
| \(L(2)\) |
\(\approx\) |
\(4.220380292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.220380292\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.11T + 8T^{2} \) |
| 5 | \( 1 + 9.56T + 125T^{2} \) |
| 7 | \( 1 + 24.0T + 343T^{2} \) |
| 13 | \( 1 - 69.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 37.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 7.31T + 1.03e5T^{2} \) |
| 53 | \( 1 - 435.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 141.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 46.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 625.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 21.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 993.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 716.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 132.T + 9.12e5T^{2} \) |
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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
−8.835078309828650300637015482853, −7.922038091828542236715908886925, −6.89457509472512226503862668333, −6.41246165439747223153749527362, −5.54319105299348532424746029232, −4.44973679671835331838010401258, −3.79887595806976690125249573706, −3.22840427034295128132716238021, −2.46487990674177837356724469583, −0.71646430175952924676516696184,
0.71646430175952924676516696184, 2.46487990674177837356724469583, 3.22840427034295128132716238021, 3.79887595806976690125249573706, 4.44973679671835331838010401258, 5.54319105299348532424746029232, 6.41246165439747223153749527362, 6.89457509472512226503862668333, 7.922038091828542236715908886925, 8.835078309828650300637015482853
