| L(s) = 1 | + 3.46·2-s − 3·3-s + 4.01·4-s − 0.957·5-s − 10.3·6-s − 1.03·7-s − 13.8·8-s + 9·9-s − 3.32·10-s + 11·11-s − 12.0·12-s + 47.3·13-s − 3.58·14-s + 2.87·15-s − 79.9·16-s + 14.2·17-s + 31.1·18-s − 40.3·19-s − 3.84·20-s + 3.09·21-s + 38.1·22-s + 76.3·23-s + 41.4·24-s − 124.·25-s + 164.·26-s − 27·27-s − 4.15·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.502·4-s − 0.0856·5-s − 0.707·6-s − 0.0557·7-s − 0.610·8-s + 0.333·9-s − 0.105·10-s + 0.301·11-s − 0.289·12-s + 1.01·13-s − 0.0683·14-s + 0.0494·15-s − 1.24·16-s + 0.203·17-s + 0.408·18-s − 0.487·19-s − 0.0430·20-s + 0.0322·21-s + 0.369·22-s + 0.691·23-s + 0.352·24-s − 0.992·25-s + 1.23·26-s − 0.192·27-s − 0.0280·28-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 | \( 1 - 3.46T + 8T^{2} \) |
| 5 | \( 1 + 0.957T + 125T^{2} \) |
| 7 | \( 1 + 1.03T + 343T^{2} \) |
| 13 | \( 1 - 47.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 76.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 37.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 19.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 15.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 474.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 521.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 856.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.15e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 480.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 386.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 728.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.67e3T + 9.12e5T^{2} \) |
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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
−8.403030228299761384424112338995, −7.37057622637308257283225832608, −6.41302112357265536313595510327, −5.94870676702369058743762007732, −5.18498597220399746413663413141, −4.25534117410043824731924312644, −3.72880064607339886491985101892, −2.67479007461989794771787219699, −1.33106502683385590658136070996, 0,
1.33106502683385590658136070996, 2.67479007461989794771787219699, 3.72880064607339886491985101892, 4.25534117410043824731924312644, 5.18498597220399746413663413141, 5.94870676702369058743762007732, 6.41302112357265536313595510327, 7.37057622637308257283225832608, 8.403030228299761384424112338995
