| L(s) = 1 | − 2-s + 2.30·3-s + 4-s − 5-s − 2.30·6-s + 0.697·7-s − 8-s + 2.30·9-s + 10-s + 2.60·11-s + 2.30·12-s + 6.30·13-s − 0.697·14-s − 2.30·15-s + 16-s − 3.90·17-s − 2.30·18-s − 0.605·19-s − 20-s + 1.60·21-s − 2.60·22-s + 1.69·23-s − 2.30·24-s + 25-s − 6.30·26-s − 1.60·27-s + 0.697·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.447·5-s − 0.940·6-s + 0.263·7-s − 0.353·8-s + 0.767·9-s + 0.316·10-s + 0.785·11-s + 0.664·12-s + 1.74·13-s − 0.186·14-s − 0.594·15-s + 0.250·16-s − 0.947·17-s − 0.542·18-s − 0.138·19-s − 0.223·20-s + 0.350·21-s − 0.555·22-s + 0.353·23-s − 0.470·24-s + 0.200·25-s − 1.23·26-s − 0.308·27-s + 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422952272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422952272\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 9.81T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 + 0.908T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 + 8.60T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−11.44715470203241819617330928940, −10.93880015048394908629159940051, −9.535443640693436142856681132436, −8.765082493453845884582129301720, −8.319202498085562194570074724910, −7.25526374973238832119327326179, −6.15149921353648408473937998277, −4.17540590010277769479068710498, −3.17916472240691115447459842518, −1.64499485444658532592961724782,
1.64499485444658532592961724782, 3.17916472240691115447459842518, 4.17540590010277769479068710498, 6.15149921353648408473937998277, 7.25526374973238832119327326179, 8.319202498085562194570074724910, 8.765082493453845884582129301720, 9.535443640693436142856681132436, 10.93880015048394908629159940051, 11.44715470203241819617330928940
