| L(s) = 1 | + 4·7-s + 11-s + 4·13-s − 4·19-s − 6·23-s + 6·29-s + 8·31-s − 2·37-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s − 6·53-s + 12·59-s + 2·61-s + 10·67-s + 12·71-s + 16·73-s + 4·77-s + 8·79-s − 6·89-s + 16·91-s − 14·97-s + 18·101-s − 14·103-s − 12·107-s − 10·109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s + 1.10·13-s − 0.917·19-s − 1.25·23-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 1.22·67-s + 1.42·71-s + 1.87·73-s + 0.455·77-s + 0.900·79-s − 0.635·89-s + 1.67·91-s − 1.42·97-s + 1.79·101-s − 1.37·103-s − 1.16·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.854447782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854447782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.097597377267241225553931888902, −6.76055372500977797865337964211, −6.48106968370125128537007638283, −5.53335716940639953879383355317, −4.89173485452934720683307316064, −4.19950469043589388497233688589, −3.60123938437220748757741756309, −2.39867365028149736702673059601, −1.71368407885028382051981284275, −0.846924886891199475929096160509,
0.846924886891199475929096160509, 1.71368407885028382051981284275, 2.39867365028149736702673059601, 3.60123938437220748757741756309, 4.19950469043589388497233688589, 4.89173485452934720683307316064, 5.53335716940639953879383355317, 6.48106968370125128537007638283, 6.76055372500977797865337964211, 8.097597377267241225553931888902
