| L(s) = 1 | − 2-s − 0.222·3-s + 4-s + 0.181·5-s + 0.222·6-s + 7-s − 8-s − 2.95·9-s − 0.181·10-s − 1.69·11-s − 0.222·12-s − 0.764·13-s − 14-s − 0.0403·15-s + 16-s − 3.26·17-s + 2.95·18-s − 0.180·19-s + 0.181·20-s − 0.222·21-s + 1.69·22-s − 9.07·23-s + 0.222·24-s − 4.96·25-s + 0.764·26-s + 1.32·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.128·3-s + 0.5·4-s + 0.0811·5-s + 0.0907·6-s + 0.377·7-s − 0.353·8-s − 0.983·9-s − 0.0573·10-s − 0.509·11-s − 0.0641·12-s − 0.212·13-s − 0.267·14-s − 0.0104·15-s + 0.250·16-s − 0.791·17-s + 0.695·18-s − 0.0414·19-s + 0.0405·20-s − 0.0484·21-s + 0.360·22-s − 1.89·23-s + 0.0453·24-s − 0.993·25-s + 0.149·26-s + 0.254·27-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
| \(L(1)\) |
\(\approx\) |
\(0.7369255555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7369255555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.222T + 3T^{2} \) |
| 5 | \( 1 - 0.181T + 5T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 0.764T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 - 2.26T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.872T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 8.43T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 - 3.51T + 71T^{2} \) |
| 73 | \( 1 + 0.619T + 73T^{2} \) |
| 79 | \( 1 - 8.64T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 0.954T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.158638425702667176602210782445, −7.61629236476940794774897224837, −6.62836466987604011586072220525, −6.05172325034511674450520994534, −5.35740356696992069928959077541, −4.49545397630793451815313547550, −3.53216311628790423804945438743, −2.47501022977333317827613260183, −1.94124944503496942076751940679, −0.47804012234695902577694261660,
0.47804012234695902577694261660, 1.94124944503496942076751940679, 2.47501022977333317827613260183, 3.53216311628790423804945438743, 4.49545397630793451815313547550, 5.35740356696992069928959077541, 6.05172325034511674450520994534, 6.62836466987604011586072220525, 7.61629236476940794774897224837, 8.158638425702667176602210782445
