| L(s) = 1 | + 2.46·2-s + 2.61·3-s + 4.06·4-s − 2.85·5-s + 6.42·6-s + 5.07·8-s + 3.81·9-s − 7.03·10-s + 3.40·11-s + 10.5·12-s + 2.37·13-s − 7.45·15-s + 4.36·16-s − 5.67·17-s + 9.39·18-s + 7.43·19-s − 11.5·20-s + 8.37·22-s + 1.38·23-s + 13.2·24-s + 3.15·25-s + 5.83·26-s + 2.12·27-s − 5.08·29-s − 18.3·30-s + 10.4·31-s + 0.603·32-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 1.50·3-s + 2.03·4-s − 1.27·5-s + 2.62·6-s + 1.79·8-s + 1.27·9-s − 2.22·10-s + 1.02·11-s + 3.05·12-s + 0.657·13-s − 1.92·15-s + 1.09·16-s − 1.37·17-s + 2.21·18-s + 1.70·19-s − 2.59·20-s + 1.78·22-s + 0.289·23-s + 2.70·24-s + 0.631·25-s + 1.14·26-s + 0.409·27-s − 0.943·29-s − 3.35·30-s + 1.88·31-s + 0.106·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.119829825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.119829825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 7.43T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 0.480T + 37T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 8.73T + 61T^{2} \) |
| 67 | \( 1 - 3.63T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−8.912883136483731849907594088366, −8.294358979568807219282642747340, −7.38226827916242485292218953698, −6.89003199248854327203263301145, −5.89934677894344706723399982250, −4.59285962091987366508824869305, −4.14547066535292531461403505095, −3.36073440338766203535736809719, −2.92223695151754569030306293850, −1.60724675219611734186585741350,
1.60724675219611734186585741350, 2.92223695151754569030306293850, 3.36073440338766203535736809719, 4.14547066535292531461403505095, 4.59285962091987366508824869305, 5.89934677894344706723399982250, 6.89003199248854327203263301145, 7.38226827916242485292218953698, 8.294358979568807219282642747340, 8.912883136483731849907594088366
