| L(s) = 1 | + 0.948·2-s − 1.10·4-s − 2.51·7-s − 2.94·8-s − 5.50·11-s − 6.20·13-s − 2.38·14-s − 0.588·16-s − 7.40·17-s + 1.48·19-s − 5.22·22-s − 7.77·23-s − 5.88·26-s + 2.76·28-s + 1.89·29-s + 31-s + 5.32·32-s − 7.02·34-s − 5.17·37-s + 1.41·38-s − 1.13·41-s + 8.20·43-s + 6.06·44-s − 7.37·46-s − 9.67·47-s − 0.688·49-s + 6.82·52-s + ⋯ |
| L(s) = 1 | + 0.670·2-s − 0.550·4-s − 0.949·7-s − 1.03·8-s − 1.66·11-s − 1.71·13-s − 0.636·14-s − 0.147·16-s − 1.79·17-s + 0.341·19-s − 1.11·22-s − 1.62·23-s − 1.15·26-s + 0.522·28-s + 0.352·29-s + 0.179·31-s + 0.941·32-s − 1.20·34-s − 0.850·37-s + 0.228·38-s − 0.177·41-s + 1.25·43-s + 0.913·44-s − 1.08·46-s − 1.41·47-s − 0.0983·49-s + 0.946·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07084945390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07084945390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.948T + 2T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 + 6.20T + 13T^{2} \) |
| 17 | \( 1 + 7.40T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 - 8.20T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 - 7.40T + 53T^{2} \) |
| 59 | \( 1 + 0.762T + 59T^{2} \) |
| 61 | \( 1 - 3.02T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| 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
−7.938722706870351912114091580748, −7.16545532091380563451273845570, −6.44871965457612054675932429281, −5.69071021227189958048191447201, −5.00718745131970675137257174728, −4.50819441030101658366201822978, −3.65327809774362855240364195329, −2.68700746611193981983307448152, −2.31026685260171066443956710632, −0.11090962940758203292306584518,
0.11090962940758203292306584518, 2.31026685260171066443956710632, 2.68700746611193981983307448152, 3.65327809774362855240364195329, 4.50819441030101658366201822978, 5.00718745131970675137257174728, 5.69071021227189958048191447201, 6.44871965457612054675932429281, 7.16545532091380563451273845570, 7.938722706870351912114091580748
