| L(s) = 1 | + 2-s + 2.12·3-s + 4-s − 4.38·5-s + 2.12·6-s + 4.48·7-s + 8-s + 1.53·9-s − 4.38·10-s − 4.38·11-s + 2.12·12-s − 4.60·13-s + 4.48·14-s − 9.33·15-s + 16-s + 2.76·17-s + 1.53·18-s + 5.48·19-s − 4.38·20-s + 9.54·21-s − 4.38·22-s − 23-s + 2.12·24-s + 14.2·25-s − 4.60·26-s − 3.12·27-s + 4.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.22·3-s + 0.5·4-s − 1.96·5-s + 0.869·6-s + 1.69·7-s + 0.353·8-s + 0.510·9-s − 1.38·10-s − 1.32·11-s + 0.614·12-s − 1.27·13-s + 1.19·14-s − 2.41·15-s + 0.250·16-s + 0.670·17-s + 0.361·18-s + 1.25·19-s − 0.980·20-s + 2.08·21-s − 0.934·22-s − 0.208·23-s + 0.434·24-s + 2.84·25-s − 0.902·26-s − 0.601·27-s + 0.846·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.811073943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.811073943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 + 4.38T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 + 4.38T + 11T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 9.33T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 - 1.93T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 9.83T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 5.85T + 61T^{2} \) |
| 67 | \( 1 + 0.491T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 - 6.24T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−7.81220122649565461330127883541, −7.59912671150455884401077863142, −7.27891482763597257656844119184, −5.54704126539246832676778135552, −4.97451245007243153637303134453, −4.36532034191243951900069057474, −3.68880992653329168383303249611, −2.81463071338727629254360523692, −2.35067480422669788140861352540, −0.868162392856792598956981337617,
0.868162392856792598956981337617, 2.35067480422669788140861352540, 2.81463071338727629254360523692, 3.68880992653329168383303249611, 4.36532034191243951900069057474, 4.97451245007243153637303134453, 5.54704126539246832676778135552, 7.27891482763597257656844119184, 7.59912671150455884401077863142, 7.81220122649565461330127883541
