| L(s) = 1 | − 9.42·3-s + 0.575·5-s + 24.5·7-s + 61.8·9-s − 11·11-s − 20.8·13-s − 5.42·15-s − 99.6·17-s − 104.·19-s − 231.·21-s + 172.·23-s − 124.·25-s − 328.·27-s + 137.·29-s − 205.·31-s + 103.·33-s + 14.1·35-s + 57.2·37-s + 196.·39-s − 247.·41-s − 353.·43-s + 35.5·45-s − 198.·47-s + 259.·49-s + 939.·51-s − 101.·53-s − 6.33·55-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 0.0514·5-s + 1.32·7-s + 2.28·9-s − 0.301·11-s − 0.444·13-s − 0.0933·15-s − 1.42·17-s − 1.26·19-s − 2.40·21-s + 1.56·23-s − 0.997·25-s − 2.33·27-s + 0.882·29-s − 1.19·31-s + 0.546·33-s + 0.0682·35-s + 0.254·37-s + 0.806·39-s − 0.943·41-s − 1.25·43-s + 0.117·45-s − 0.616·47-s + 0.756·49-s + 2.57·51-s − 0.263·53-s − 0.0155·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 9.42T + 27T^{2} \) |
| 5 | \( 1 - 0.575T + 125T^{2} \) |
| 7 | \( 1 - 24.5T + 343T^{2} \) |
| 13 | \( 1 + 20.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 57.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 247.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 101.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 240.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 43.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 631.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 38.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 388.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 119.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 520.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 335.T + 9.12e5T^{2} \) |
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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
−11.43217869182875523207981948163, −11.08395063548935231372371712384, −10.17639037290031666053766102362, −8.649260457315214967808471289650, −7.28445255266712673261291904227, −6.32316700916703020744668050793, −5.08249721555386345128294042862, −4.52029281861718636569270372321, −1.75077660292205735339352452610, 0,
1.75077660292205735339352452610, 4.52029281861718636569270372321, 5.08249721555386345128294042862, 6.32316700916703020744668050793, 7.28445255266712673261291904227, 8.649260457315214967808471289650, 10.17639037290031666053766102362, 11.08395063548935231372371712384, 11.43217869182875523207981948163
