| L(s) = 1 | + 2.94·3-s + 5-s + 7-s + 5.66·9-s + 4.38·11-s − 13-s + 2.94·15-s + 1.63·17-s − 4.89·19-s + 2.94·21-s + 3.49·23-s + 25-s + 7.83·27-s − 4.44·29-s + 4.94·31-s + 12.9·33-s + 35-s + 5.58·37-s − 2.94·39-s − 10.5·41-s − 7.79·43-s + 5.66·45-s + 1.74·47-s + 49-s + 4.80·51-s + 5.11·53-s + 4.38·55-s + ⋯ |
| L(s) = 1 | + 1.69·3-s + 0.447·5-s + 0.377·7-s + 1.88·9-s + 1.32·11-s − 0.277·13-s + 0.759·15-s + 0.396·17-s − 1.12·19-s + 0.642·21-s + 0.729·23-s + 0.200·25-s + 1.50·27-s − 0.824·29-s + 0.887·31-s + 2.24·33-s + 0.169·35-s + 0.918·37-s − 0.471·39-s − 1.64·41-s − 1.18·43-s + 0.844·45-s + 0.254·47-s + 0.142·49-s + 0.673·51-s + 0.701·53-s + 0.591·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.486028484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.486028484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 - 9.03T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 - 0.113T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 0.922T + 97T^{2} \) |
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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
−8.421826705477492074395386333623, −8.176353791038457397687105646639, −7.02320140215926971872881960337, −6.67057944804865629513738101804, −5.47876130500583589073642276062, −4.43246530483727542576762129353, −3.81176174038894279555907304011, −2.94025072514370220462051411074, −2.07761501300982015155491535568, −1.31575819138935286516122964210,
1.31575819138935286516122964210, 2.07761501300982015155491535568, 2.94025072514370220462051411074, 3.81176174038894279555907304011, 4.43246530483727542576762129353, 5.47876130500583589073642276062, 6.67057944804865629513738101804, 7.02320140215926971872881960337, 8.176353791038457397687105646639, 8.421826705477492074395386333623
