| L(s) = 1 | + 0.00424·2-s − 7.83·3-s − 7.99·4-s + 8.54·5-s − 0.0332·6-s + 10.0·7-s − 0.0678·8-s + 34.4·9-s + 0.0362·10-s + 34.4·11-s + 62.6·12-s + 6.17·13-s + 0.0424·14-s − 66.9·15-s + 63.9·16-s + 122.·17-s + 0.146·18-s − 71.8·19-s − 68.3·20-s − 78.4·21-s + 0.145·22-s − 96.2·23-s + 0.532·24-s − 51.9·25-s + 0.0261·26-s − 58.1·27-s − 80.0·28-s + ⋯ |
| L(s) = 1 | + 0.00150·2-s − 1.50·3-s − 0.999·4-s + 0.764·5-s − 0.00226·6-s + 0.540·7-s − 0.00300·8-s + 1.27·9-s + 0.00114·10-s + 0.943·11-s + 1.50·12-s + 0.131·13-s + 0.000810·14-s − 1.15·15-s + 0.999·16-s + 1.75·17-s + 0.00191·18-s − 0.867·19-s − 0.764·20-s − 0.814·21-s + 0.00141·22-s − 0.873·23-s + 0.00452·24-s − 0.415·25-s + 0.000197·26-s − 0.414·27-s − 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9159667249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9159667249\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 + 83T \) |
| good | 2 | \( 1 - 0.00424T + 8T^{2} \) |
| 3 | \( 1 + 7.83T + 27T^{2} \) |
| 5 | \( 1 - 8.54T + 125T^{2} \) |
| 7 | \( 1 - 10.0T + 343T^{2} \) |
| 11 | \( 1 - 34.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.17T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 555.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 565.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 601.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 582.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 431.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 284.T + 4.93e5T^{2} \) |
| 89 | \( 1 + 606.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3T + 9.12e5T^{2} \) |
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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
−13.83940784132543229885243053512, −12.50031829112448036627984628060, −11.79481114176635270845823304606, −10.43446250717173365139230047021, −9.655979292618060917288505351900, −8.163304746605992155649711334268, −6.30868507868186638118546797977, −5.47647278092553750727631189901, −4.27926978109631419220073308917, −1.05101575795431504811394056301,
1.05101575795431504811394056301, 4.27926978109631419220073308917, 5.47647278092553750727631189901, 6.30868507868186638118546797977, 8.163304746605992155649711334268, 9.655979292618060917288505351900, 10.43446250717173365139230047021, 11.79481114176635270845823304606, 12.50031829112448036627984628060, 13.83940784132543229885243053512
