| L(s) = 1 | − 5·5-s + 22.6·7-s + 10.9·11-s − 75.8·13-s + 15.9·17-s + 58.8·19-s − 56.6·23-s + 25·25-s − 5.61·29-s − 120.·31-s − 113.·35-s + 236.·37-s − 196.·41-s + 18.1·43-s − 311.·47-s + 168.·49-s − 33.3·53-s − 54.8·55-s − 520.·59-s − 150.·61-s + 379.·65-s + 506.·67-s − 961.·71-s − 251.·73-s + 248.·77-s + 834.·79-s − 691.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.22·7-s + 0.300·11-s − 1.61·13-s + 0.227·17-s + 0.710·19-s − 0.513·23-s + 0.200·25-s − 0.0359·29-s − 0.699·31-s − 0.546·35-s + 1.04·37-s − 0.747·41-s + 0.0645·43-s − 0.967·47-s + 0.491·49-s − 0.0865·53-s − 0.134·55-s − 1.14·59-s − 0.315·61-s + 0.723·65-s + 0.922·67-s − 1.60·71-s − 0.403·73-s + 0.367·77-s + 1.18·79-s − 0.914·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 22.6T + 343T^{2} \) |
| 11 | \( 1 - 10.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 5.61T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 18.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 33.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 520.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 150.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 506.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 834.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 691.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−8.536255243932489232299101423176, −7.64751099539347360182011096350, −7.40337939400384712522990627070, −6.18192516045412447018682638887, −5.06707593762914655019432007190, −4.65653980370144075440780076885, −3.53667356053239389889186281641, −2.38333245819842615631281939975, −1.35804736935083042376826899860, 0,
1.35804736935083042376826899860, 2.38333245819842615631281939975, 3.53667356053239389889186281641, 4.65653980370144075440780076885, 5.06707593762914655019432007190, 6.18192516045412447018682638887, 7.40337939400384712522990627070, 7.64751099539347360182011096350, 8.536255243932489232299101423176
