L(s) = 1 | − 2.04·2-s − 3-s + 2.18·4-s − 1.44·5-s + 2.04·6-s − 4.07·7-s − 0.369·8-s + 9-s + 2.95·10-s + 11-s − 2.18·12-s − 0.740·13-s + 8.32·14-s + 1.44·15-s − 3.60·16-s − 5.60·17-s − 2.04·18-s + 7.10·19-s − 3.15·20-s + 4.07·21-s − 2.04·22-s + 6.31·23-s + 0.369·24-s − 2.90·25-s + 1.51·26-s − 27-s − 8.87·28-s + ⋯ |
L(s) = 1 | − 1.44·2-s − 0.577·3-s + 1.09·4-s − 0.647·5-s + 0.834·6-s − 1.53·7-s − 0.130·8-s + 0.333·9-s + 0.935·10-s + 0.301·11-s − 0.629·12-s − 0.205·13-s + 2.22·14-s + 0.373·15-s − 0.901·16-s − 1.35·17-s − 0.481·18-s + 1.62·19-s − 0.705·20-s + 0.888·21-s − 0.435·22-s + 1.31·23-s + 0.0754·24-s − 0.581·25-s + 0.297·26-s − 0.192·27-s − 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 13 | \( 1 + 0.740T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 - 9.71T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 67 | \( 1 - 7.40T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−9.109879570013266660886538648668, −8.015567018777939552051293982339, −7.11708921031933230395591752962, −6.85550492043540250628562864677, −5.85518764744762858952232152636, −4.67761854920569487375285694882, −3.65275218246625698282978143269, −2.57100234094703296651475495025, −0.999133069216803901406858833985, 0,
0.999133069216803901406858833985, 2.57100234094703296651475495025, 3.65275218246625698282978143269, 4.67761854920569487375285694882, 5.85518764744762858952232152636, 6.85550492043540250628562864677, 7.11708921031933230395591752962, 8.015567018777939552051293982339, 9.109879570013266660886538648668