| L(s) = 1 | + 2-s − 0.823·3-s + 4-s + 1.30·5-s − 0.823·6-s + 0.990·7-s + 8-s − 2.32·9-s + 1.30·10-s + 2.45·11-s − 0.823·12-s − 1.10·13-s + 0.990·14-s − 1.07·15-s + 16-s + 3.67·17-s − 2.32·18-s + 19-s + 1.30·20-s − 0.815·21-s + 2.45·22-s + 3.34·23-s − 0.823·24-s − 3.30·25-s − 1.10·26-s + 4.38·27-s + 0.990·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.475·3-s + 0.5·4-s + 0.582·5-s − 0.336·6-s + 0.374·7-s + 0.353·8-s − 0.774·9-s + 0.411·10-s + 0.740·11-s − 0.237·12-s − 0.306·13-s + 0.264·14-s − 0.276·15-s + 0.250·16-s + 0.891·17-s − 0.547·18-s + 0.229·19-s + 0.291·20-s − 0.177·21-s + 0.523·22-s + 0.697·23-s − 0.168·24-s − 0.661·25-s − 0.216·26-s + 0.843·27-s + 0.187·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.448303512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.448303512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 0.823T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 - 0.990T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 3.67T + 17T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 7.61T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 9.71T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 9.49T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 1.69T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.759T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.99T + 89T^{2} \) |
| 97 | \( 1 - 4.59T + 97T^{2} \) |
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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
−7.79174976834489943377295636855, −6.82825365601739177290302698518, −6.33249315309504144208295615205, −5.73574677892516630601838025144, −5.00779134490208197810515199090, −4.57946056150197827900760929660, −3.41232407931511725528043801476, −2.85029552734381152579490067410, −1.81077583287134859950782308056, −0.877088655494500926222283933983,
0.877088655494500926222283933983, 1.81077583287134859950782308056, 2.85029552734381152579490067410, 3.41232407931511725528043801476, 4.57946056150197827900760929660, 5.00779134490208197810515199090, 5.73574677892516630601838025144, 6.33249315309504144208295615205, 6.82825365601739177290302698518, 7.79174976834489943377295636855
