| L(s) = 1 | − 0.823·5-s + 0.910·7-s + 3.82·13-s − 0.564·17-s − 5.25·19-s + 3.18·23-s − 4.32·25-s + 7.92·29-s + 5.46·31-s − 0.750·35-s − 2.00·37-s − 1.17·41-s + 4.07·43-s + 2.84·47-s − 6.17·49-s + 6.35·53-s − 5.49·59-s − 3.45·61-s − 3.15·65-s − 3.14·67-s + 12.4·71-s − 8.93·73-s − 0.443·79-s + 8.22·83-s + 0.464·85-s − 6.50·89-s + 3.48·91-s + ⋯ |
| L(s) = 1 | − 0.368·5-s + 0.344·7-s + 1.06·13-s − 0.136·17-s − 1.20·19-s + 0.664·23-s − 0.864·25-s + 1.47·29-s + 0.981·31-s − 0.126·35-s − 0.329·37-s − 0.183·41-s + 0.621·43-s + 0.415·47-s − 0.881·49-s + 0.873·53-s − 0.715·59-s − 0.442·61-s − 0.391·65-s − 0.383·67-s + 1.47·71-s − 1.04·73-s − 0.0499·79-s + 0.903·83-s + 0.0504·85-s − 0.689·89-s + 0.365·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.974345916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.974345916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 0.823T + 5T^{2} \) |
| 7 | \( 1 - 0.910T + 7T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 0.564T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 4.07T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 + 5.49T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 8.93T + 73T^{2} \) |
| 79 | \( 1 + 0.443T + 79T^{2} \) |
| 83 | \( 1 - 8.22T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + 1.44T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.906345347177522405564574658737, −7.03627583896229576934247953755, −6.37266415126675960781555912319, −5.84024985435289232558367227674, −4.80837625497281728294368679888, −4.31123256888013015430496568840, −3.52807997283133567853383320606, −2.66827952784046611443105102586, −1.70565076617796200774763910350, −0.70035419350998391865103147059,
0.70035419350998391865103147059, 1.70565076617796200774763910350, 2.66827952784046611443105102586, 3.52807997283133567853383320606, 4.31123256888013015430496568840, 4.80837625497281728294368679888, 5.84024985435289232558367227674, 6.37266415126675960781555912319, 7.03627583896229576934247953755, 7.906345347177522405564574658737
