| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 11-s + 14-s − 16-s − 2·17-s + 4·19-s + 22-s + 8·23-s − 5·25-s − 28-s + 4·29-s + 8·31-s + 5·32-s − 2·34-s + 8·37-s + 4·38-s − 2·41-s − 12·43-s − 44-s + 8·46-s − 2·47-s + 49-s − 5·50-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 0.301·11-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.213·22-s + 1.66·23-s − 25-s − 0.188·28-s + 0.742·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s + 1.31·37-s + 0.648·38-s − 0.312·41-s − 1.82·43-s − 0.150·44-s + 1.17·46-s − 0.291·47-s + 1/7·49-s − 0.707·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.73414368590654, −13.33491019636365, −13.14504723297497, −12.32991592191042, −12.00506251629638, −11.41884755566059, −11.25797447231462, −10.32113241287272, −9.942016678823500, −9.397704266961281, −8.947314359703878, −8.450285806117831, −7.932370948085806, −7.397569191160982, −6.638650098629811, −6.284820345714818, −5.717282082996134, −4.958321217806539, −4.768428162459644, −4.277405791001505, −3.474934839096536, −3.062008206517180, −2.511867523975941, −1.495064819785432, −0.9414627671630904, 0,
0.9414627671630904, 1.495064819785432, 2.511867523975941, 3.062008206517180, 3.474934839096536, 4.277405791001505, 4.768428162459644, 4.958321217806539, 5.717282082996134, 6.284820345714818, 6.638650098629811, 7.397569191160982, 7.932370948085806, 8.450285806117831, 8.947314359703878, 9.397704266961281, 9.942016678823500, 10.32113241287272, 11.25797447231462, 11.41884755566059, 12.00506251629638, 12.32991592191042, 13.14504723297497, 13.33491019636365, 13.73414368590654
