| L(s) = 1 | + 3.15·3-s − 3.03·5-s + 7.94·7-s − 17.0·9-s + 27.6·11-s − 58.1·13-s − 9.56·15-s − 17·17-s + 89.1·19-s + 25.0·21-s + 115.·23-s − 115.·25-s − 138.·27-s + 128.·29-s − 273.·31-s + 87.1·33-s − 24.0·35-s + 132.·37-s − 183.·39-s − 470.·41-s + 352.·43-s + 51.6·45-s − 152.·47-s − 279.·49-s − 53.6·51-s − 527.·53-s − 83.7·55-s + ⋯ |
| L(s) = 1 | + 0.607·3-s − 0.271·5-s + 0.428·7-s − 0.631·9-s + 0.756·11-s − 1.23·13-s − 0.164·15-s − 0.242·17-s + 1.07·19-s + 0.260·21-s + 1.04·23-s − 0.926·25-s − 0.990·27-s + 0.823·29-s − 1.58·31-s + 0.459·33-s − 0.116·35-s + 0.588·37-s − 0.752·39-s − 1.79·41-s + 1.25·43-s + 0.171·45-s − 0.473·47-s − 0.816·49-s − 0.147·51-s − 1.36·53-s − 0.205·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.15T + 27T^{2} \) |
| 5 | \( 1 + 3.03T + 125T^{2} \) |
| 7 | \( 1 - 7.94T + 343T^{2} \) |
| 11 | \( 1 - 27.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 89.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 527.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 53.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 52.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 295.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 720.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 813.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 794.T + 9.12e5T^{2} \) |
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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
−9.228540015967689179983324594780, −8.183565538694189315563801843063, −7.57182699920063468669007875805, −6.71890057304727234289822296833, −5.52159601964081746460923003238, −4.71763662487752261318024018994, −3.57992051928312276859738526788, −2.72377054766116607072981342075, −1.55210243232429648271510253485, 0,
1.55210243232429648271510253485, 2.72377054766116607072981342075, 3.57992051928312276859738526788, 4.71763662487752261318024018994, 5.52159601964081746460923003238, 6.71890057304727234289822296833, 7.57182699920063468669007875805, 8.183565538694189315563801843063, 9.228540015967689179983324594780
