| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s + 2·13-s + 16-s + 6·17-s + 18-s + 4·19-s + 24-s + 2·26-s + 27-s − 6·29-s − 8·31-s + 32-s + 6·34-s + 36-s − 2·37-s + 4·38-s + 2·39-s + 6·41-s + 4·43-s + 48-s + 6·51-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.144·48-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.699296103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.699296103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.66810840475203951668812462194, −7.37246722336349349577457464296, −6.48483889073410329581405140915, −5.46615702258852877719138323154, −5.38202845134137609952360890459, −4.04294344334526682970159599548, −3.64669715609605890343404486151, −2.90048846214809368117965694867, −1.94564685631743918385494852366, −1.00883045224893318465988836621,
1.00883045224893318465988836621, 1.94564685631743918385494852366, 2.90048846214809368117965694867, 3.64669715609605890343404486151, 4.04294344334526682970159599548, 5.38202845134137609952360890459, 5.46615702258852877719138323154, 6.48483889073410329581405140915, 7.37246722336349349577457464296, 7.66810840475203951668812462194
