| L(s) = 1 | + 9·3-s − 77.8·5-s + 81·9-s + 58.2·11-s + 792.·13-s − 700.·15-s − 464.·17-s − 984.·19-s − 635.·23-s + 2.92e3·25-s + 729·27-s + 6.39e3·29-s − 8.42e3·31-s + 524.·33-s − 1.40e3·37-s + 7.13e3·39-s + 1.90e4·41-s + 6.02e3·43-s − 6.30e3·45-s + 1.88e3·47-s − 4.18e3·51-s + 5.25e3·53-s − 4.53e3·55-s − 8.86e3·57-s + 3.11e3·59-s − 2.20e4·61-s − 6.16e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.39·5-s + 0.333·9-s + 0.145·11-s + 1.30·13-s − 0.803·15-s − 0.389·17-s − 0.625·19-s − 0.250·23-s + 0.937·25-s + 0.192·27-s + 1.41·29-s − 1.57·31-s + 0.0838·33-s − 0.168·37-s + 0.750·39-s + 1.77·41-s + 0.496·43-s − 0.463·45-s + 0.124·47-s − 0.225·51-s + 0.256·53-s − 0.202·55-s − 0.361·57-s + 0.116·59-s − 0.760·61-s − 1.81·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 77.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 58.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 792.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 464.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 984.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 635.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.90e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.25e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.11e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.47e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.14e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.22e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.32e5T + 8.58e9T^{2} \) |
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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
−9.179824552167691254763182508250, −8.548556505604708824298826885505, −7.82709361751945199894632629829, −6.97936386262207675745046950479, −5.91215797365971355005207897090, −4.36454756566770340642516640536, −3.86844594198730166535112330847, −2.79213489696167184403304112950, −1.30805452537466530198294503437, 0,
1.30805452537466530198294503437, 2.79213489696167184403304112950, 3.86844594198730166535112330847, 4.36454756566770340642516640536, 5.91215797365971355005207897090, 6.97936386262207675745046950479, 7.82709361751945199894632629829, 8.548556505604708824298826885505, 9.179824552167691254763182508250
