| L(s) = 1 | + 2-s + 4-s − 3.30·7-s + 8-s + 5.34·11-s − 5.34·13-s − 3.30·14-s + 16-s − 1.23·17-s + 2.38·19-s + 5.34·22-s + 5.85·23-s − 5.34·26-s − 3.30·28-s + 6.60·29-s + 10.4·31-s + 32-s − 1.23·34-s + 3.30·37-s + 2.38·38-s − 7.38·41-s + 10.6·43-s + 5.34·44-s + 5.85·46-s − 3.09·47-s + 3.90·49-s − 5.34·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.24·7-s + 0.353·8-s + 1.61·11-s − 1.48·13-s − 0.882·14-s + 0.250·16-s − 0.299·17-s + 0.546·19-s + 1.13·22-s + 1.22·23-s − 1.04·26-s − 0.624·28-s + 1.22·29-s + 1.88·31-s + 0.176·32-s − 0.211·34-s + 0.543·37-s + 0.386·38-s − 1.15·41-s + 1.63·43-s + 0.805·44-s + 0.863·46-s − 0.450·47-s + 0.558·49-s − 0.741·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.608513336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.608513336\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 - 4.08T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.276343281714208623780051811520, −8.232425258705726765917828830124, −6.99484532353494687902932575732, −6.79154161465344472294036024591, −5.99039004647521339077296222427, −4.90783795984354038666388865297, −4.22442743258303149534040658092, −3.20170191994318645603667872750, −2.54979719062032121946869467550, −0.968354880577502114916302992359,
0.968354880577502114916302992359, 2.54979719062032121946869467550, 3.20170191994318645603667872750, 4.22442743258303149534040658092, 4.90783795984354038666388865297, 5.99039004647521339077296222427, 6.79154161465344472294036024591, 6.99484532353494687902932575732, 8.232425258705726765917828830124, 9.276343281714208623780051811520
