| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 6·13-s + 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 4·22-s − 24-s − 25-s − 6·26-s + 27-s − 29-s − 2·30-s − 4·31-s − 32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.185·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138222379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138222379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.08350822284660829527532209241, −11.46160956762346043433851090643, −10.51228751975807998983720994202, −9.664379188026696343396565275536, −8.703567572426077992686024733312, −7.83419862053560271569394084997, −6.51962651722825426517927501982, −5.35523044786598344694970550055, −3.32512902126658354016325680833, −1.81977103661582150142638563575,
1.81977103661582150142638563575, 3.32512902126658354016325680833, 5.35523044786598344694970550055, 6.51962651722825426517927501982, 7.83419862053560271569394084997, 8.703567572426077992686024733312, 9.664379188026696343396565275536, 10.51228751975807998983720994202, 11.46160956762346043433851090643, 13.08350822284660829527532209241
