| L(s) = 1 | + 5.35·2-s + 3·3-s + 20.6·4-s + 12.4·5-s + 16.0·6-s − 25.2·7-s + 67.8·8-s + 9·9-s + 66.6·10-s + 11·11-s + 62.0·12-s + 19.9·13-s − 135.·14-s + 37.3·15-s + 197.·16-s − 9.87·17-s + 48.1·18-s + 19.0·19-s + 257.·20-s − 75.8·21-s + 58.8·22-s + 174.·23-s + 203.·24-s + 29.8·25-s + 106.·26-s + 27·27-s − 522.·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.58·4-s + 1.11·5-s + 1.09·6-s − 1.36·7-s + 2.99·8-s + 0.333·9-s + 2.10·10-s + 0.301·11-s + 1.49·12-s + 0.424·13-s − 2.58·14-s + 0.642·15-s + 3.09·16-s − 0.140·17-s + 0.631·18-s + 0.229·19-s + 2.87·20-s − 0.788·21-s + 0.570·22-s + 1.58·23-s + 1.73·24-s + 0.238·25-s + 0.804·26-s + 0.192·27-s − 3.52·28-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
| \(L(2)\) |
\(\approx\) |
\(12.03527404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.03527404\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.35T + 8T^{2} \) |
| 5 | \( 1 - 12.4T + 125T^{2} \) |
| 7 | \( 1 + 25.2T + 343T^{2} \) |
| 13 | \( 1 - 19.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.87T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 25.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 145.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 26.6T + 2.05e5T^{2} \) |
| 67 | \( 1 + 817.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 104.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 738.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 880.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 89.0T + 9.12e5T^{2} \) |
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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
−9.052972621440311836980554500273, −7.56673947008134378135809412718, −6.85152276817264694180260705097, −6.16927036341523703795665378630, −5.70178492585757129590499217347, −4.69144263156648305436342526941, −3.75557845234251627126402747229, −3.03351140978053535858316702123, −2.41169392573592983346178243818, −1.29368689649395138004498649550,
1.29368689649395138004498649550, 2.41169392573592983346178243818, 3.03351140978053535858316702123, 3.75557845234251627126402747229, 4.69144263156648305436342526941, 5.70178492585757129590499217347, 6.16927036341523703795665378630, 6.85152276817264694180260705097, 7.56673947008134378135809412718, 9.052972621440311836980554500273
