| L(s) = 1 | − 0.653·2-s + 1.14·3-s − 1.57·4-s + 4.22·5-s − 0.745·6-s + 2.33·8-s − 1.69·9-s − 2.76·10-s − 4.06·11-s − 1.79·12-s − 4.00·13-s + 4.82·15-s + 1.61·16-s − 1.96·17-s + 1.10·18-s + 3.59·19-s − 6.64·20-s + 2.65·22-s + 8.25·23-s + 2.66·24-s + 12.8·25-s + 2.61·26-s − 5.36·27-s − 0.680·29-s − 3.15·30-s + 9.32·31-s − 5.72·32-s + ⋯ |
| L(s) = 1 | − 0.462·2-s + 0.658·3-s − 0.786·4-s + 1.89·5-s − 0.304·6-s + 0.825·8-s − 0.565·9-s − 0.873·10-s − 1.22·11-s − 0.518·12-s − 1.11·13-s + 1.24·15-s + 0.404·16-s − 0.475·17-s + 0.261·18-s + 0.825·19-s − 1.48·20-s + 0.566·22-s + 1.72·23-s + 0.543·24-s + 2.57·25-s + 0.513·26-s − 1.03·27-s − 0.126·29-s − 0.575·30-s + 1.67·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.965437412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965437412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.653T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + 4.00T + 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 - 8.25T + 23T^{2} \) |
| 29 | \( 1 + 0.680T + 29T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 9.18T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
| 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
−8.352482020942943917734006075665, −7.43350192866700952323992701013, −6.77939408111254211112105450593, −5.71369665116526634980286594268, −5.10162953622582298064631634941, −4.85206826902044412724048218817, −3.27455197150054802613286496715, −2.62656101238978848511185560979, −1.97351749655966014658314648876, −0.75574082341319068424358181169,
0.75574082341319068424358181169, 1.97351749655966014658314648876, 2.62656101238978848511185560979, 3.27455197150054802613286496715, 4.85206826902044412724048218817, 5.10162953622582298064631634941, 5.71369665116526634980286594268, 6.77939408111254211112105450593, 7.43350192866700952323992701013, 8.352482020942943917734006075665
