| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−11.04696060108841953029776373543, −10.34115200803620225763799917905, −8.924195929553904985738525965411, −8.230423205343344515355480256609, −6.86376739299499030341353505875, −5.81710138455725822863304685397, −5.03757430943605731463273866715, −3.50439886027380408681771841824, −1.85328112104661425664183214095, 0,
1.85328112104661425664183214095, 3.50439886027380408681771841824, 5.03757430943605731463273866715, 5.81710138455725822863304685397, 6.86376739299499030341353505875, 8.230423205343344515355480256609, 8.924195929553904985738525965411, 10.34115200803620225763799917905, 11.04696060108841953029776373543
