| L(s) = 1 | − 0.607·2-s + 3·3-s − 7.63·4-s + 5·5-s − 1.82·6-s + 8.95·7-s + 9.49·8-s + 9·9-s − 3.03·10-s − 11·11-s − 22.8·12-s − 0.460·13-s − 5.43·14-s + 15·15-s + 55.2·16-s + 128.·17-s − 5.46·18-s − 0.0245·19-s − 38.1·20-s + 26.8·21-s + 6.67·22-s + 171.·23-s + 28.4·24-s + 25·25-s + 0.279·26-s + 27·27-s − 68.3·28-s + ⋯ |
| L(s) = 1 | − 0.214·2-s + 0.577·3-s − 0.953·4-s + 0.447·5-s − 0.123·6-s + 0.483·7-s + 0.419·8-s + 0.333·9-s − 0.0960·10-s − 0.301·11-s − 0.550·12-s − 0.00982·13-s − 0.103·14-s + 0.258·15-s + 0.863·16-s + 1.83·17-s − 0.0715·18-s − 0.000296·19-s − 0.426·20-s + 0.279·21-s + 0.0647·22-s + 1.55·23-s + 0.242·24-s + 0.200·25-s + 0.00210·26-s + 0.192·27-s − 0.461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.765210952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765210952\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.607T + 8T^{2} \) |
| 7 | \( 1 - 8.95T + 343T^{2} \) |
| 13 | \( 1 + 0.460T + 2.19e3T^{2} \) |
| 17 | \( 1 - 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.0245T + 6.85e3T^{2} \) |
| 23 | \( 1 - 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 622.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 9.86T + 2.05e5T^{2} \) |
| 61 | \( 1 - 902.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 146.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 459.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 125.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−12.73571293142000328895449976337, −11.24903380010802816436380458651, −10.00212734113146905023655586956, −9.388079821420692639793492986243, −8.277180068951443661438294092648, −7.50139287119856281710070420224, −5.69102629869008333045958502837, −4.61749048961360917943542706486, −3.12791487994090885759162751116, −1.20747782620850776850538122683,
1.20747782620850776850538122683, 3.12791487994090885759162751116, 4.61749048961360917943542706486, 5.69102629869008333045958502837, 7.50139287119856281710070420224, 8.277180068951443661438294092648, 9.388079821420692639793492986243, 10.00212734113146905023655586956, 11.24903380010802816436380458651, 12.73571293142000328895449976337
