| L(s) = 1 | + 16·7-s + 60·11-s − 86·13-s + 18·17-s + 44·19-s + 48·23-s + 186·29-s + 176·31-s − 254·37-s − 186·41-s + 100·43-s + 168·47-s − 87·49-s − 498·53-s + 252·59-s − 58·61-s + 1.03e3·67-s − 168·71-s − 506·73-s + 960·77-s + 272·79-s + 948·83-s + 1.01e3·89-s − 1.37e3·91-s + 766·97-s + 1.31e3·101-s + 448·103-s + ⋯ |
| L(s) = 1 | + 0.863·7-s + 1.64·11-s − 1.83·13-s + 0.256·17-s + 0.531·19-s + 0.435·23-s + 1.19·29-s + 1.01·31-s − 1.12·37-s − 0.708·41-s + 0.354·43-s + 0.521·47-s − 0.253·49-s − 1.29·53-s + 0.556·59-s − 0.121·61-s + 1.88·67-s − 0.280·71-s − 0.811·73-s + 1.42·77-s + 0.387·79-s + 1.25·83-s + 1.20·89-s − 1.58·91-s + 0.801·97-s + 1.29·101-s + 0.428·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.468978347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.468978347\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 186 T + p^{3} T^{2} \) |
| 43 | \( 1 - 100 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 506 T + p^{3} T^{2} \) |
| 79 | \( 1 - 272 T + p^{3} T^{2} \) |
| 83 | \( 1 - 948 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1014 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−9.702991241570410780171358128809, −8.939883697051137491842518360183, −8.020825724973308430833228349938, −7.15978178683215796729606996749, −6.42008588730990409307368726667, −5.09869533240505731708023140056, −4.55649022780536756304313673175, −3.30219497183459181297020827740, −2.03858748360329180999120560866, −0.895978897670521456684068744556,
0.895978897670521456684068744556, 2.03858748360329180999120560866, 3.30219497183459181297020827740, 4.55649022780536756304313673175, 5.09869533240505731708023140056, 6.42008588730990409307368726667, 7.15978178683215796729606996749, 8.020825724973308430833228349938, 8.939883697051137491842518360183, 9.702991241570410780171358128809
