| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 2.94·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 5.88·10-s + 33.3·11-s + 12·12-s + 71.9·13-s − 14·14-s + 8.83·15-s + 16·16-s + 53.4·17-s + 18·18-s − 75.3·19-s + 11.7·20-s − 21·21-s + 66.6·22-s − 23·23-s + 24·24-s − 116.·25-s + 143.·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.263·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.186·10-s + 0.912·11-s + 0.288·12-s + 1.53·13-s − 0.267·14-s + 0.152·15-s + 0.250·16-s + 0.762·17-s + 0.235·18-s − 0.910·19-s + 0.131·20-s − 0.218·21-s + 0.645·22-s − 0.208·23-s + 0.204·24-s − 0.930·25-s + 1.08·26-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
| \(L(2)\) |
\(\approx\) |
\(4.909410576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.909410576\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.94T + 125T^{2} \) |
| 11 | \( 1 - 33.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 76.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 54.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 289.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 36.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 904.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 898.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 516.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 662.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 100.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 82.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 295.T + 9.12e5T^{2} \) |
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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
−9.737478973373597838955082360631, −8.651159877168772794810328116436, −8.116609245543295800919503615558, −6.75234535245433842247916977332, −6.32774804813803949087285927592, −5.29954920792350624403736005125, −3.95621049510524309332352766599, −3.55704694465401901791333655579, −2.24853368874607931973785792396, −1.13825742169368656890469222182,
1.13825742169368656890469222182, 2.24853368874607931973785792396, 3.55704694465401901791333655579, 3.95621049510524309332352766599, 5.29954920792350624403736005125, 6.32774804813803949087285927592, 6.75234535245433842247916977332, 8.116609245543295800919503615558, 8.651159877168772794810328116436, 9.737478973373597838955082360631
