| L(s) = 1 | − 5.31·2-s + 20.2·4-s + 7.25·5-s + 7·7-s − 64.8·8-s − 38.5·10-s − 11·11-s + 47.6·13-s − 37.1·14-s + 182.·16-s + 31.5·17-s + 18.9·19-s + 146.·20-s + 58.4·22-s − 200.·23-s − 72.4·25-s − 252.·26-s + 141.·28-s − 224.·29-s − 237.·31-s − 451.·32-s − 167.·34-s + 50.7·35-s + 226.·37-s − 100.·38-s − 470.·40-s + 31.1·41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.52·4-s + 0.648·5-s + 0.377·7-s − 2.86·8-s − 1.21·10-s − 0.301·11-s + 1.01·13-s − 0.709·14-s + 2.85·16-s + 0.450·17-s + 0.228·19-s + 1.63·20-s + 0.566·22-s − 1.81·23-s − 0.579·25-s − 1.90·26-s + 0.954·28-s − 1.43·29-s − 1.37·31-s − 2.49·32-s − 0.846·34-s + 0.245·35-s + 1.00·37-s − 0.429·38-s − 1.85·40-s + 0.118·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.31T + 8T^{2} \) |
| 5 | \( 1 - 7.25T + 125T^{2} \) |
| 13 | \( 1 - 47.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 18.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 226.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 526.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 283.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 216.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 180.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 166.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 44.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 349.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 722.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.80e3T + 9.12e5T^{2} \) |
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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
−9.706850439499811078214885934235, −8.812944544786879543990530630336, −8.029806349263941816948544719183, −7.41380892590641811667438112025, −6.22145331493700210900972562143, −5.61573829162751410666521351545, −3.64985400309496618666967521102, −2.16997981498564748434833471902, −1.44908673800444283445717491115, 0,
1.44908673800444283445717491115, 2.16997981498564748434833471902, 3.64985400309496618666967521102, 5.61573829162751410666521351545, 6.22145331493700210900972562143, 7.41380892590641811667438112025, 8.029806349263941816948544719183, 8.812944544786879543990530630336, 9.706850439499811078214885934235
