| L(s) = 1 | − 2-s + 3.22·3-s + 4-s + 1.18·5-s − 3.22·6-s + 1.12·7-s − 8-s + 7.41·9-s − 1.18·10-s − 3.41·11-s + 3.22·12-s + 0.347·13-s − 1.12·14-s + 3.82·15-s + 16-s − 7.41·18-s + 0.347·19-s + 1.18·20-s + 3.61·21-s + 3.41·22-s − 0.411·23-s − 3.22·24-s − 3.59·25-s − 0.347·26-s + 14.2·27-s + 1.12·28-s − 8.78·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s + 0.529·5-s − 1.31·6-s + 0.423·7-s − 0.353·8-s + 2.47·9-s − 0.374·10-s − 1.02·11-s + 0.931·12-s + 0.0963·13-s − 0.299·14-s + 0.987·15-s + 0.250·16-s − 1.74·18-s + 0.0796·19-s + 0.264·20-s + 0.789·21-s + 0.727·22-s − 0.0857·23-s − 0.658·24-s − 0.719·25-s − 0.0681·26-s + 2.73·27-s + 0.211·28-s − 1.63·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162435121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162435121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 0.347T + 13T^{2} \) |
| 19 | \( 1 - 0.347T + 19T^{2} \) |
| 23 | \( 1 + 0.411T + 23T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 - 8.71T + 31T^{2} \) |
| 37 | \( 1 + 0.475T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 + 7.86T + 47T^{2} \) |
| 53 | \( 1 - 8.41T + 53T^{2} \) |
| 59 | \( 1 + 6.41T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 - 9.04T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 7.73T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 - 0.221T + 97T^{2} \) |
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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
−10.22930138279139119731815534077, −9.731928029928731678294533603328, −8.935862368630885126871719829128, −8.018927480707213954673114043477, −7.74161040975547886945699768277, −6.52556597470936298670291265460, −5.05287546227699737645502250541, −3.64686034098190402595071896663, −2.56050764196916455916413239937, −1.72349135090505258963251456535,
1.72349135090505258963251456535, 2.56050764196916455916413239937, 3.64686034098190402595071896663, 5.05287546227699737645502250541, 6.52556597470936298670291265460, 7.74161040975547886945699768277, 8.018927480707213954673114043477, 8.935862368630885126871719829128, 9.731928029928731678294533603328, 10.22930138279139119731815534077
