| L(s) = 1 | + 1.43·2-s − 3-s + 0.0567·4-s + 4.14·5-s − 1.43·6-s + 7-s − 2.78·8-s + 9-s + 5.94·10-s − 3.13·11-s − 0.0567·12-s + 1.25·13-s + 1.43·14-s − 4.14·15-s − 4.11·16-s + 1.72·17-s + 1.43·18-s − 3.13·19-s + 0.235·20-s − 21-s − 4.50·22-s − 2.47·23-s + 2.78·24-s + 12.1·25-s + 1.79·26-s − 27-s + 0.0567·28-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.577·3-s + 0.0283·4-s + 1.85·5-s − 0.585·6-s + 0.377·7-s − 0.985·8-s + 0.333·9-s + 1.87·10-s − 0.946·11-s − 0.0163·12-s + 0.347·13-s + 0.383·14-s − 1.06·15-s − 1.02·16-s + 0.418·17-s + 0.338·18-s − 0.719·19-s + 0.0525·20-s − 0.218·21-s − 0.959·22-s − 0.516·23-s + 0.568·24-s + 2.43·25-s + 0.352·26-s − 0.192·27-s + 0.0107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377441101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377441101\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 11 | \( 1 + 3.13T + 11T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 5.74T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 9.16T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 + 8.31T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 1.19T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 + 6.96T + 89T^{2} \) |
| 97 | \( 1 - 8.15T + 97T^{2} \) |
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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
−8.528571726305293860632627865409, −7.58967692276047608951640960970, −6.46150570275809442934366853759, −5.90707572631880518980945160640, −5.62764351000877784917571655992, −4.78319661237952542017358325212, −4.19798124749280849867359926361, −2.81095434967450838634609888679, −2.26203811331724284323543304018, −0.958040797592371900947023952235,
0.958040797592371900947023952235, 2.26203811331724284323543304018, 2.81095434967450838634609888679, 4.19798124749280849867359926361, 4.78319661237952542017358325212, 5.62764351000877784917571655992, 5.90707572631880518980945160640, 6.46150570275809442934366853759, 7.58967692276047608951640960970, 8.528571726305293860632627865409
