| L(s) = 1 | − 5-s + 3·7-s − 5·11-s − 2·13-s + 17-s − 19-s + 4·23-s − 4·25-s + 6·29-s + 10·31-s − 3·35-s + 11·43-s + 9·47-s + 2·49-s − 10·53-s + 5·55-s + 4·59-s − 5·61-s + 2·65-s + 4·67-s + 8·71-s + 13·73-s − 15·77-s − 4·79-s − 4·83-s − 85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 1.50·11-s − 0.554·13-s + 0.242·17-s − 0.229·19-s + 0.834·23-s − 4/5·25-s + 1.11·29-s + 1.79·31-s − 0.507·35-s + 1.67·43-s + 1.31·47-s + 2/7·49-s − 1.37·53-s + 0.674·55-s + 0.520·59-s − 0.640·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s + 1.52·73-s − 1.70·77-s − 0.450·79-s − 0.439·83-s − 0.108·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657407710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657407710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.603714387546405910501403215999, −7.88268435972744262251626047055, −7.68229866741704553480915382531, −6.60362314557230975715150598611, −5.55340583692568326567045997877, −4.86969759310695125905179309034, −4.29030743174506950540377994277, −2.96552278342726800960012222910, −2.22993996642987601042956119625, −0.799584790040422697199795345098,
0.799584790040422697199795345098, 2.22993996642987601042956119625, 2.96552278342726800960012222910, 4.29030743174506950540377994277, 4.86969759310695125905179309034, 5.55340583692568326567045997877, 6.60362314557230975715150598611, 7.68229866741704553480915382531, 7.88268435972744262251626047055, 8.603714387546405910501403215999
