| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 13-s − 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s + 4·22-s + 8·23-s + 24-s + 25-s − 26-s + 27-s + 6·29-s − 30-s + 8·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.355221917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.355221917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.58916795151074, −15.02549840117790, −14.60718458264148, −14.07660889049047, −13.80172937097296, −12.84437998632699, −12.48309903718480, −12.02419287087664, −11.47662401429625, −10.72646441834635, −10.26686729165134, −9.496851361836205, −8.951811863174233, −8.311277970467514, −7.761561502142575, −7.014588861728314, −6.617882144658443, −5.925074964587748, −5.019332361194754, −4.514696456141756, −3.866259270550518, −3.159460641303711, −2.722906501352728, −1.580684169830841, −0.9039134974644530,
0.9039134974644530, 1.580684169830841, 2.722906501352728, 3.159460641303711, 3.866259270550518, 4.514696456141756, 5.019332361194754, 5.925074964587748, 6.617882144658443, 7.014588861728314, 7.761561502142575, 8.311277970467514, 8.951811863174233, 9.496851361836205, 10.26686729165134, 10.72646441834635, 11.47662401429625, 12.02419287087664, 12.48309903718480, 12.84437998632699, 13.80172937097296, 14.07660889049047, 14.60718458264148, 15.02549840117790, 15.58916795151074
