| L(s) = 1 | + 2-s − 1.69·3-s + 4-s − 2.70·5-s − 1.69·6-s − 7-s + 8-s − 0.133·9-s − 2.70·10-s + 6.10·11-s − 1.69·12-s + 6.88·13-s − 14-s + 4.57·15-s + 16-s − 2.56·17-s − 0.133·18-s + 4.90·19-s − 2.70·20-s + 1.69·21-s + 6.10·22-s − 0.388·23-s − 1.69·24-s + 2.29·25-s + 6.88·26-s + 5.30·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.977·3-s + 0.5·4-s − 1.20·5-s − 0.691·6-s − 0.377·7-s + 0.353·8-s − 0.0445·9-s − 0.854·10-s + 1.84·11-s − 0.488·12-s + 1.91·13-s − 0.267·14-s + 1.18·15-s + 0.250·16-s − 0.622·17-s − 0.0315·18-s + 1.12·19-s − 0.603·20-s + 0.369·21-s + 1.30·22-s − 0.0810·23-s − 0.345·24-s + 0.458·25-s + 1.35·26-s + 1.02·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.805110298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.805110298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.69T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 - 6.88T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 + 0.388T + 23T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 + 0.566T + 59T^{2} \) |
| 61 | \( 1 - 4.12T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 8.29T + 73T^{2} \) |
| 79 | \( 1 + 0.750T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.927660327717070528702893964491, −7.06301700916183739287430775983, −6.51953143291595008369581881143, −5.99359375041597707246843793476, −5.27330599625221440456730367805, −4.30350352926161044184940349186, −3.64207251614351495255882361596, −3.38771522100920921503200153433, −1.66197670972582521187086552962, −0.69923385968867606370723325712,
0.69923385968867606370723325712, 1.66197670972582521187086552962, 3.38771522100920921503200153433, 3.64207251614351495255882361596, 4.30350352926161044184940349186, 5.27330599625221440456730367805, 5.99359375041597707246843793476, 6.51953143291595008369581881143, 7.06301700916183739287430775983, 7.927660327717070528702893964491
