L(s) = 1 | + 3.47·2-s + 4.08·4-s + 7.60·5-s + 16.3·7-s − 13.6·8-s + 26.4·10-s + 42.4·11-s + 62.2·13-s + 56.9·14-s − 79.9·16-s − 17·17-s − 52.0·19-s + 31.0·20-s + 147.·22-s + 67.8·23-s − 67.1·25-s + 216.·26-s + 67.0·28-s + 91.7·29-s − 177.·31-s − 169.·32-s − 59.1·34-s + 124.·35-s − 258.·37-s − 181.·38-s − 103.·40-s + 262.·41-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.511·4-s + 0.680·5-s + 0.885·7-s − 0.601·8-s + 0.836·10-s + 1.16·11-s + 1.32·13-s + 1.08·14-s − 1.24·16-s − 0.242·17-s − 0.629·19-s + 0.347·20-s + 1.42·22-s + 0.615·23-s − 0.537·25-s + 1.63·26-s + 0.452·28-s + 0.587·29-s − 1.02·31-s − 0.935·32-s − 0.298·34-s + 0.602·35-s − 1.14·37-s − 0.773·38-s − 0.408·40-s + 0.998·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.553574526\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.553574526\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 + 17T \) |
good | 2 | \( 1 - 3.47T + 8T^{2} \) |
| 5 | \( 1 - 7.60T + 125T^{2} \) |
| 7 | \( 1 - 16.3T + 343T^{2} \) |
| 11 | \( 1 - 42.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 67.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 262.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 331.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 49.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 41.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 845.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 402.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 463.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−12.74011078774311245884991315754, −11.65159954884225246772236373896, −10.88481840607633758307109417790, −9.323358673628956548676796627596, −8.467239196079302647020900765164, −6.67228043362126840036621343648, −5.82407569020152945177409053080, −4.65342662234711925541295199523, −3.55213125816418111606059672348, −1.70701678348170145307387506065,
1.70701678348170145307387506065, 3.55213125816418111606059672348, 4.65342662234711925541295199523, 5.82407569020152945177409053080, 6.67228043362126840036621343648, 8.467239196079302647020900765164, 9.323358673628956548676796627596, 10.88481840607633758307109417790, 11.65159954884225246772236373896, 12.74011078774311245884991315754