| L(s) = 1 | + 1.38·2-s + 3-s − 0.0806·4-s + 1.70·5-s + 1.38·6-s − 4.16·7-s − 2.88·8-s + 9-s + 2.36·10-s + 5.40·11-s − 0.0806·12-s − 5.67·13-s − 5.76·14-s + 1.70·15-s − 3.83·16-s − 17-s + 1.38·18-s + 3.65·19-s − 0.137·20-s − 4.16·21-s + 7.49·22-s + 5.25·23-s − 2.88·24-s − 2.08·25-s − 7.85·26-s + 27-s + 0.335·28-s + ⋯ |
| L(s) = 1 | + 0.979·2-s + 0.577·3-s − 0.0403·4-s + 0.763·5-s + 0.565·6-s − 1.57·7-s − 1.01·8-s + 0.333·9-s + 0.748·10-s + 1.63·11-s − 0.0232·12-s − 1.57·13-s − 1.54·14-s + 0.441·15-s − 0.958·16-s − 0.242·17-s + 0.326·18-s + 0.839·19-s − 0.0307·20-s − 0.908·21-s + 1.59·22-s + 1.09·23-s − 0.588·24-s − 0.416·25-s − 1.54·26-s + 0.192·27-s + 0.0634·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.423615092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.423615092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 - 5.40T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 + 8.21T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 0.252T + 43T^{2} \) |
| 47 | \( 1 + 5.84T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 2.97T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
| 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
−7.56254969399923832923298092642, −6.91874139342577259870126128026, −6.38839731601133080887340866344, −5.74017412143329373463981445628, −5.00485923640900177402814520855, −4.14864673967673241586924769224, −3.55273998337857274865519442688, −2.87122622624569039063875435113, −2.18860751324831304696899342827, −0.75004396907952985092878754885,
0.75004396907952985092878754885, 2.18860751324831304696899342827, 2.87122622624569039063875435113, 3.55273998337857274865519442688, 4.14864673967673241586924769224, 5.00485923640900177402814520855, 5.74017412143329373463981445628, 6.38839731601133080887340866344, 6.91874139342577259870126128026, 7.56254969399923832923298092642
