| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 4·11-s − 12-s + 2·13-s + 14-s + 16-s + 3·17-s − 18-s + 21-s + 4·22-s − 23-s + 24-s − 5·25-s − 2·26-s − 27-s − 28-s + 9·29-s + 3·31-s − 32-s + 4·33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.218·21-s + 0.852·22-s − 0.208·23-s + 0.204·24-s − 25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.696·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.66929909523627, −12.25300170896700, −11.79022063885369, −11.42779078549152, −10.86440340086975, −10.38536486040812, −10.05672987632526, −9.872496736232138, −9.151372540109514, −8.587554513210003, −8.162737434734809, −7.865758586325690, −7.219943403183133, −6.823662434567265, −6.287565448402202, −5.833333866632199, −5.371044618948228, −4.963256954804999, −4.178712424717787, −3.731704765426342, −3.016714261025505, −2.589600133824995, −1.969670153678634, −1.205749057593357, −0.6837753285200477, 0,
0.6837753285200477, 1.205749057593357, 1.969670153678634, 2.589600133824995, 3.016714261025505, 3.731704765426342, 4.178712424717787, 4.963256954804999, 5.371044618948228, 5.833333866632199, 6.287565448402202, 6.823662434567265, 7.219943403183133, 7.865758586325690, 8.162737434734809, 8.587554513210003, 9.151372540109514, 9.872496736232138, 10.05672987632526, 10.38536486040812, 10.86440340086975, 11.42779078549152, 11.79022063885369, 12.25300170896700, 12.66929909523627
