| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 3·7-s + 8-s + 9-s − 5·11-s + 12-s + 2·13-s − 3·14-s + 16-s + 7·17-s + 18-s − 2·19-s − 3·21-s − 5·22-s − 4·23-s + 24-s + 2·26-s + 27-s − 3·28-s − 5·29-s − 7·31-s + 32-s − 5·33-s + 7·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.554·13-s − 0.801·14-s + 1/4·16-s + 1.69·17-s + 0.235·18-s − 0.458·19-s − 0.654·21-s − 1.06·22-s − 0.834·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.566·28-s − 0.928·29-s − 1.25·31-s + 0.176·32-s − 0.870·33-s + 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.86470367650231189327077125456, −7.07792457963267676996370643248, −6.23644218844457452497738036995, −5.61656279809980960172029664351, −4.96322111450714942722128840093, −3.74706000204562754436940603909, −3.40245969538060390507733787870, −2.62529616085175060156116484578, −1.64878512607014007983199363993, 0,
1.64878512607014007983199363993, 2.62529616085175060156116484578, 3.40245969538060390507733787870, 3.74706000204562754436940603909, 4.96322111450714942722128840093, 5.61656279809980960172029664351, 6.23644218844457452497738036995, 7.07792457963267676996370643248, 7.86470367650231189327077125456
