| L(s) = 1 | + 1.16·2-s + 1.13·3-s − 0.644·4-s − 1.45·5-s + 1.31·6-s + 3.35·7-s − 3.07·8-s − 1.71·9-s − 1.69·10-s − 1.76·11-s − 0.729·12-s − 2.24·13-s + 3.91·14-s − 1.65·15-s − 2.29·16-s + 2.23·17-s − 1.99·18-s + 6.09·19-s + 0.939·20-s + 3.80·21-s − 2.05·22-s − 3.34·23-s − 3.48·24-s − 2.87·25-s − 2.61·26-s − 5.34·27-s − 2.16·28-s + ⋯ |
| L(s) = 1 | + 0.823·2-s + 0.653·3-s − 0.322·4-s − 0.651·5-s + 0.538·6-s + 1.26·7-s − 1.08·8-s − 0.572·9-s − 0.536·10-s − 0.531·11-s − 0.210·12-s − 0.622·13-s + 1.04·14-s − 0.426·15-s − 0.573·16-s + 0.540·17-s − 0.471·18-s + 1.39·19-s + 0.210·20-s + 0.830·21-s − 0.437·22-s − 0.696·23-s − 0.711·24-s − 0.575·25-s − 0.512·26-s − 1.02·27-s − 0.409·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363214317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363214317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 - T \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 - 9.55T + 29T^{2} \) |
| 31 | \( 1 - 0.417T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 6.20T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 + 0.00636T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 + 5.78T + 73T^{2} \) |
| 79 | \( 1 + 5.78T + 79T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−14.28169387582182941518301595891, −13.59394730722657302197008318326, −12.12659726636073883281381181809, −11.51737086901092806935943944472, −9.801042849625701485378150919948, −8.400273899238101564825174331551, −7.73892480725517251556952315078, −5.55978851979933749921956786088, −4.47231178755777228337340260814, −2.98080145095345604661409664966,
2.98080145095345604661409664966, 4.47231178755777228337340260814, 5.55978851979933749921956786088, 7.73892480725517251556952315078, 8.400273899238101564825174331551, 9.801042849625701485378150919948, 11.51737086901092806935943944472, 12.12659726636073883281381181809, 13.59394730722657302197008318326, 14.28169387582182941518301595891
