| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 17.2·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 34.4·10-s + 63.6·11-s + 12·12-s + 63.3·13-s − 14·14-s + 51.7·15-s + 16·16-s − 41.6·17-s − 18·18-s + 58.1·19-s + 68.9·20-s + 21·21-s − 127.·22-s − 23·23-s − 24·24-s + 172.·25-s − 126.·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.54·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.08·10-s + 1.74·11-s + 0.288·12-s + 1.35·13-s − 0.267·14-s + 0.889·15-s + 0.250·16-s − 0.594·17-s − 0.235·18-s + 0.702·19-s + 0.770·20-s + 0.218·21-s − 1.23·22-s − 0.208·23-s − 0.204·24-s + 1.37·25-s − 0.956·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.352156216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.352156216\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 17.2T + 125T^{2} \) |
| 11 | \( 1 - 63.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 31.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 548.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 246.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 327.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 743.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 263.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 265.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 490.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.21e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 422.T + 9.12e5T^{2} \) |
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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
−9.592325295218721401660661004202, −8.790215560591690927595024557547, −8.413658197208438423526329989355, −6.95201278876435153369721281761, −6.42634728819765187071397075842, −5.55472277245648676224331851170, −4.16800758932508140241741593157, −3.00940971035387296102811327378, −1.69841063017908614306712528005, −1.28524603757798497608824609137,
1.28524603757798497608824609137, 1.69841063017908614306712528005, 3.00940971035387296102811327378, 4.16800758932508140241741593157, 5.55472277245648676224331851170, 6.42634728819765187071397075842, 6.95201278876435153369721281761, 8.413658197208438423526329989355, 8.790215560591690927595024557547, 9.592325295218721401660661004202
