| L(s) = 1 | − 2-s − 0.725·3-s + 4-s + 3.71·5-s + 0.725·6-s − 1.04·7-s − 8-s − 2.47·9-s − 3.71·10-s + 0.736·11-s − 0.725·12-s − 5.03·13-s + 1.04·14-s − 2.69·15-s + 16-s + 3.26·17-s + 2.47·18-s − 5.55·19-s + 3.71·20-s + 0.761·21-s − 0.736·22-s + 8.81·23-s + 0.725·24-s + 8.83·25-s + 5.03·26-s + 3.97·27-s − 1.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.419·3-s + 0.5·4-s + 1.66·5-s + 0.296·6-s − 0.396·7-s − 0.353·8-s − 0.824·9-s − 1.17·10-s + 0.222·11-s − 0.209·12-s − 1.39·13-s + 0.280·14-s − 0.697·15-s + 0.250·16-s + 0.791·17-s + 0.582·18-s − 1.27·19-s + 0.831·20-s + 0.166·21-s − 0.157·22-s + 1.83·23-s + 0.148·24-s + 1.76·25-s + 0.986·26-s + 0.764·27-s − 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232006340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232006340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 0.725T + 3T^{2} \) |
| 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 0.736T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 - 8.81T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 - 6.41T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 - 1.98T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 - 5.30T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 9.32T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
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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
−9.449282809838192990982255447823, −8.989071100849292370971941473998, −7.993543328445662323597394962002, −6.82381501566652022246959964473, −6.33701496148691770728830734233, −5.53325197334634816986558423152, −4.77542018767270738530548943477, −2.90386178377329835422528881602, −2.33728314531846865359106920919, −0.894721600720139226623972685954,
0.894721600720139226623972685954, 2.33728314531846865359106920919, 2.90386178377329835422528881602, 4.77542018767270738530548943477, 5.53325197334634816986558423152, 6.33701496148691770728830734233, 6.82381501566652022246959964473, 7.993543328445662323597394962002, 8.989071100849292370971941473998, 9.449282809838192990982255447823
