| L(s) = 1 | − 0.796·2-s − 1.36·4-s − 4.63·7-s + 2.68·8-s + 4.73·11-s − 3.69·13-s + 3.69·14-s + 0.593·16-s − 3.88·17-s − 4.32·19-s − 3.76·22-s + 4.51·23-s + 2.94·26-s + 6.32·28-s + 7.07·29-s − 2.63·31-s − 5.83·32-s + 3.09·34-s + 37-s + 3.44·38-s − 8.22·41-s + 5.81·43-s − 6.45·44-s − 3.59·46-s − 0.961·47-s + 14.4·49-s + 5.03·52-s + ⋯ |
| L(s) = 1 | − 0.563·2-s − 0.682·4-s − 1.75·7-s + 0.948·8-s + 1.42·11-s − 1.02·13-s + 0.986·14-s + 0.148·16-s − 0.942·17-s − 0.991·19-s − 0.803·22-s + 0.941·23-s + 0.576·26-s + 1.19·28-s + 1.31·29-s − 0.472·31-s − 1.03·32-s + 0.530·34-s + 0.164·37-s + 0.558·38-s − 1.28·41-s + 0.887·43-s − 0.973·44-s − 0.530·46-s − 0.140·47-s + 2.06·49-s + 0.698·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 0.796T + 2T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 3.88T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 - 4.51T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 2.63T + 31T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 + 0.961T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 - 1.18T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 5.13T + 79T^{2} \) |
| 83 | \( 1 + 2.30T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.26450195224530559738245478717, −6.83184513890573569157705938803, −6.35340937722898602845258734986, −5.38483469067391235621131561127, −4.44963788136782428993345637599, −3.97791972346717325939285391213, −3.11280439213868186004323668145, −2.18678897926544661140835992745, −0.911581132442095027599122223421, 0,
0.911581132442095027599122223421, 2.18678897926544661140835992745, 3.11280439213868186004323668145, 3.97791972346717325939285391213, 4.44963788136782428993345637599, 5.38483469067391235621131561127, 6.35340937722898602845258734986, 6.83184513890573569157705938803, 7.26450195224530559738245478717
