| L(s) = 1 | − 17.1·5-s + 23.1·7-s − 11·11-s − 59.6·13-s + 80.8·17-s + 11.7·19-s − 56.0·23-s + 168.·25-s + 85.4·29-s + 99.8·31-s − 396.·35-s + 402.·37-s − 27.0·41-s + 74·43-s − 408.·47-s + 192.·49-s − 463.·53-s + 188.·55-s − 498.·59-s − 635.·61-s + 1.02e3·65-s + 701.·67-s + 27.7·71-s + 619.·73-s − 254.·77-s + 208.·79-s − 1.30e3·83-s + ⋯ |
| L(s) = 1 | − 1.53·5-s + 1.24·7-s − 0.301·11-s − 1.27·13-s + 1.15·17-s + 0.141·19-s − 0.508·23-s + 1.34·25-s + 0.547·29-s + 0.578·31-s − 1.91·35-s + 1.79·37-s − 0.103·41-s + 0.262·43-s − 1.26·47-s + 0.560·49-s − 1.20·53-s + 0.462·55-s − 1.10·59-s − 1.33·61-s + 1.95·65-s + 1.27·67-s + 0.0463·71-s + 0.993·73-s − 0.376·77-s + 0.296·79-s − 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 17.1T + 125T^{2} \) |
| 7 | \( 1 - 23.1T + 343T^{2} \) |
| 13 | \( 1 + 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 27.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74T + 7.95e4T^{2} \) |
| 47 | \( 1 + 408.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 635.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 701.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 27.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 208.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.278595329809874528341132007612, −7.78244830374909943685011737400, −7.53580027999244594433788440635, −6.26277538124417949784904758899, −4.94827193607931544864424037013, −4.65395880685323897777534853217, −3.57516889948019528588592057699, −2.55890113389633905797180988300, −1.17760953987608182624534403609, 0,
1.17760953987608182624534403609, 2.55890113389633905797180988300, 3.57516889948019528588592057699, 4.65395880685323897777534853217, 4.94827193607931544864424037013, 6.26277538124417949784904758899, 7.53580027999244594433788440635, 7.78244830374909943685011737400, 8.278595329809874528341132007612
