| L(s) = 1 | + 9.40·3-s − 11.8·5-s − 5.65·7-s + 61.4·9-s + 63.6·11-s − 7.43·13-s − 111.·15-s − 17·17-s − 82.1·19-s − 53.1·21-s + 176.·23-s + 16.1·25-s + 323.·27-s + 58.2·29-s + 294.·31-s + 598.·33-s + 67.1·35-s − 259.·37-s − 69.8·39-s − 360.·41-s + 332.·43-s − 730.·45-s − 4.62·47-s − 311.·49-s − 159.·51-s + 354.·53-s − 756.·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 1.06·5-s − 0.305·7-s + 2.27·9-s + 1.74·11-s − 0.158·13-s − 1.92·15-s − 0.242·17-s − 0.991·19-s − 0.552·21-s + 1.60·23-s + 0.129·25-s + 2.30·27-s + 0.372·29-s + 1.70·31-s + 3.15·33-s + 0.324·35-s − 1.15·37-s − 0.286·39-s − 1.37·41-s + 1.17·43-s − 2.41·45-s − 0.0143·47-s − 0.906·49-s − 0.438·51-s + 0.919·53-s − 1.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.868367862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.868367862\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 9.40T + 27T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 7 | \( 1 + 5.65T + 343T^{2} \) |
| 11 | \( 1 - 63.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.43T + 2.19e3T^{2} \) |
| 19 | \( 1 + 82.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 294.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.62T + 1.03e5T^{2} \) |
| 53 | \( 1 - 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 498.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 752.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 507.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 364.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 976.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 329.T + 9.12e5T^{2} \) |
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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
−9.253607284548276093689508102212, −8.571668487224830220670192996588, −8.155659688996650137183185896766, −6.96342572538746338661803118569, −6.69479762901121484447550347379, −4.69670417510562961213427459192, −3.89546015832827720944986847884, −3.36301798061853526555272428224, −2.26416514550180449107854572672, −0.993359859782260034562564188187,
0.993359859782260034562564188187, 2.26416514550180449107854572672, 3.36301798061853526555272428224, 3.89546015832827720944986847884, 4.69670417510562961213427459192, 6.69479762901121484447550347379, 6.96342572538746338661803118569, 8.155659688996650137183185896766, 8.571668487224830220670192996588, 9.253607284548276093689508102212
