| L(s) = 1 | − 2-s + 3-s − 4-s + 2·5-s − 6-s − 7-s + 3·8-s + 9-s − 2·10-s − 4·11-s − 12-s − 2·13-s + 14-s + 2·15-s − 16-s − 18-s − 4·19-s − 2·20-s − 21-s + 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s + 28-s − 29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962328895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962328895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.15022606845981, −12.85585456477992, −12.57552911975625, −11.68486065896849, −11.12087995291830, −10.51379660783782, −10.19414698210582, −9.758059245317198, −9.498891099257310, −8.957173476083470, −8.366598366170433, −8.061309723491639, −7.554444910099958, −7.078347173162327, −6.370321994631172, −5.839514613501463, −5.384042539282841, −4.641261764210089, −4.353831343554065, −3.667793531443752, −2.819478418368935, −2.322757776912719, −2.065048128536361, −0.9819400878984541, −0.5177292314264116,
0.5177292314264116, 0.9819400878984541, 2.065048128536361, 2.322757776912719, 2.819478418368935, 3.667793531443752, 4.353831343554065, 4.641261764210089, 5.384042539282841, 5.839514613501463, 6.370321994631172, 7.078347173162327, 7.554444910099958, 8.061309723491639, 8.366598366170433, 8.957173476083470, 9.498891099257310, 9.758059245317198, 10.19414698210582, 10.51379660783782, 11.12087995291830, 11.68486065896849, 12.57552911975625, 12.85585456477992, 13.15022606845981
