| L(s) = 1 | − 2.78·2-s + 0.298·3-s + 5.73·4-s − 5-s − 0.828·6-s − 7-s − 10.3·8-s − 2.91·9-s + 2.78·10-s + 1.93·11-s + 1.70·12-s + 0.800·13-s + 2.78·14-s − 0.298·15-s + 17.3·16-s + 0.984·17-s + 8.09·18-s − 7.86·19-s − 5.73·20-s − 0.298·21-s − 5.36·22-s + 3.94·23-s − 3.09·24-s + 25-s − 2.22·26-s − 1.76·27-s − 5.73·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.172·3-s + 2.86·4-s − 0.447·5-s − 0.338·6-s − 0.377·7-s − 3.66·8-s − 0.970·9-s + 0.879·10-s + 0.581·11-s + 0.493·12-s + 0.222·13-s + 0.743·14-s − 0.0769·15-s + 4.34·16-s + 0.238·17-s + 1.90·18-s − 1.80·19-s − 1.28·20-s − 0.0650·21-s − 1.14·22-s + 0.823·23-s − 0.631·24-s + 0.200·25-s − 0.436·26-s − 0.339·27-s − 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3292216631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3292216631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 - 0.298T + 3T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 - 0.800T + 13T^{2} \) |
| 17 | \( 1 - 0.984T + 17T^{2} \) |
| 19 | \( 1 + 7.86T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 - 9.35T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 6.96T + 43T^{2} \) |
| 47 | \( 1 + 8.50T + 47T^{2} \) |
| 53 | \( 1 - 5.63T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 5.62T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.087916620232317462755097628509, −7.36613812008255300641897115978, −6.64393229850429118362117564709, −6.20846796488618580953867420855, −5.38410996826688576867642325998, −3.92120748868386615145927242350, −3.19516091361839445177554244994, −2.38622632426659982240865998090, −1.56187700067941348457286497940, −0.37390346748649307617534102714,
0.37390346748649307617534102714, 1.56187700067941348457286497940, 2.38622632426659982240865998090, 3.19516091361839445177554244994, 3.92120748868386615145927242350, 5.38410996826688576867642325998, 6.20846796488618580953867420855, 6.64393229850429118362117564709, 7.36613812008255300641897115978, 8.087916620232317462755097628509
