| L(s) = 1 | − 3.40·7-s − 4.40·11-s + 2.06·13-s − 1.40·17-s − 6.35·19-s + 0.107·23-s + 9.08·29-s + 3.06·31-s + 1.95·37-s − 8.69·41-s − 7.12·43-s − 7.48·47-s + 4.57·49-s + 13.0·53-s − 13.1·59-s − 1.72·61-s − 12.3·67-s − 7.50·71-s − 5.42·73-s + 14.9·77-s + 9.43·79-s + 8.97·83-s + 4.01·89-s − 7.01·91-s + 2.17·97-s + 2.97·101-s + 6.63·103-s + ⋯ |
| L(s) = 1 | − 1.28·7-s − 1.32·11-s + 0.572·13-s − 0.339·17-s − 1.45·19-s + 0.0224·23-s + 1.68·29-s + 0.550·31-s + 0.321·37-s − 1.35·41-s − 1.08·43-s − 1.09·47-s + 0.653·49-s + 1.79·53-s − 1.71·59-s − 0.220·61-s − 1.51·67-s − 0.891·71-s − 0.634·73-s + 1.70·77-s + 1.06·79-s + 0.985·83-s + 0.425·89-s − 0.735·91-s + 0.220·97-s + 0.295·101-s + 0.653·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8683986865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8683986865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 0.107T + 23T^{2} \) |
| 29 | \( 1 - 9.08T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 - 4.01T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.949895832158813331426120754730, −6.95597219968883371568309078825, −6.42614652669087957408202447864, −5.95858240188731023888670803046, −4.94400866793047750618557508824, −4.35982124137082331071038669384, −3.28950757934832563436475636448, −2.84424498915974634648161498579, −1.87389194093565742748624513489, −0.43587311795446973631098246856,
0.43587311795446973631098246856, 1.87389194093565742748624513489, 2.84424498915974634648161498579, 3.28950757934832563436475636448, 4.35982124137082331071038669384, 4.94400866793047750618557508824, 5.95858240188731023888670803046, 6.42614652669087957408202447864, 6.95597219968883371568309078825, 7.949895832158813331426120754730
