| L(s) = 1 | − 4.41·5-s + (4.14 + 18.0i)7-s + 68.3i·11-s − 34.6i·13-s − 21.8·17-s − 124. i·19-s + 70.8i·23-s − 105.·25-s − 216. i·29-s − 280. i·31-s + (−18.2 − 79.6i)35-s + 150.·37-s + 272.·41-s + 273.·43-s + 194.·47-s + ⋯ |
| L(s) = 1 | − 0.394·5-s + (0.223 + 0.974i)7-s + 1.87i·11-s − 0.738i·13-s − 0.311·17-s − 1.50i·19-s + 0.642i·23-s − 0.844·25-s − 1.38i·29-s − 1.62i·31-s + (−0.0882 − 0.384i)35-s + 0.670·37-s + 1.03·41-s + 0.969·43-s + 0.604·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.802956315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802956315\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-4.14 - 18.0i)T \) |
| good | 5 | \( 1 + 4.41T + 125T^{2} \) |
| 11 | \( 1 - 68.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 34.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 21.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 70.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 216. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 280. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 272.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 273.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 520. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 602.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 168. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 742.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 758. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 274.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 244. iT - 9.12e5T^{2} \) |
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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
−8.741634497175270700853528638143, −7.68813074498928864154004659873, −7.46173671359082753111109243137, −6.30335500364625500208434990176, −5.52794924745351809846451393019, −4.63738511942557134824742034060, −4.01457913140275973193011467056, −2.54484888967468482689233271182, −2.14620254775287237926847614658, −0.56568723109838507396971955169,
0.65324469629281636565093552427, 1.54445049867978664328452783209, 3.00845628468831794379082788922, 3.79523182478855122482236010165, 4.42795705360033128411353528390, 5.60819186389181037991673451161, 6.27776762126959720249573807654, 7.17681020280181750624004337712, 7.904486221237981938259846464666, 8.591581410684148643523996739750
