| L(s) = 1 | + 2-s + 0.366·3-s + 4-s − 0.953·5-s + 0.366·6-s + 3.52·7-s + 8-s − 2.86·9-s − 0.953·10-s + 5.52·11-s + 0.366·12-s − 3.59·13-s + 3.52·14-s − 0.349·15-s + 16-s + 3.46·17-s − 2.86·18-s + 3.06·19-s − 0.953·20-s + 1.29·21-s + 5.52·22-s + 1.26·23-s + 0.366·24-s − 4.08·25-s − 3.59·26-s − 2.14·27-s + 3.52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.211·3-s + 0.5·4-s − 0.426·5-s + 0.149·6-s + 1.33·7-s + 0.353·8-s − 0.955·9-s − 0.301·10-s + 1.66·11-s + 0.105·12-s − 0.997·13-s + 0.943·14-s − 0.0901·15-s + 0.250·16-s + 0.841·17-s − 0.675·18-s + 0.703·19-s − 0.213·20-s + 0.281·21-s + 1.17·22-s + 0.264·23-s + 0.0747·24-s − 0.817·25-s − 0.705·26-s − 0.413·27-s + 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.475536843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.475536843\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 0.366T + 3T^{2} \) |
| 5 | \( 1 + 0.953T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 + 3.59T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 - 5.26T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 + 7.77T + 43T^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 + 1.53T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 2.33T + 79T^{2} \) |
| 83 | \( 1 + 1.34T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 0.295T + 97T^{2} \) |
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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
−11.29219800568661760041087426651, −9.978004232283093625838805744195, −8.959706719449284210213805243555, −7.988013157561947536731354827507, −7.31302662325247527211510648088, −6.05581578771803549524746274516, −5.09246513419165112829498524698, −4.16852336935295580792149910450, −3.06710570530098722488802717016, −1.58299826477557201101447556563,
1.58299826477557201101447556563, 3.06710570530098722488802717016, 4.16852336935295580792149910450, 5.09246513419165112829498524698, 6.05581578771803549524746274516, 7.31302662325247527211510648088, 7.988013157561947536731354827507, 8.959706719449284210213805243555, 9.978004232283093625838805744195, 11.29219800568661760041087426651
