| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 4·11-s + 6·13-s − 2·14-s + 16-s − 4·17-s + 19-s + 20-s − 4·22-s + 25-s − 6·26-s + 2·28-s + 10·29-s − 2·31-s − 32-s + 4·34-s + 2·35-s − 2·37-s − 38-s − 40-s − 8·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.229·19-s + 0.223·20-s − 0.852·22-s + 1/5·25-s − 1.17·26-s + 0.377·28-s + 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.338·35-s − 0.328·37-s − 0.162·38-s − 0.158·40-s − 1.24·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724784722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724784722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.081650781769527718108637078862, −8.667891024523460275253883213989, −8.044107996050708391409834608218, −6.67717098920958314152394624363, −6.51914956387136270852546336776, −5.35708181227167532787761077019, −4.30032808357235128666275695206, −3.29298652740753695701377317054, −1.89790938700513335557070325628, −1.12737143444698159586204498120,
1.12737143444698159586204498120, 1.89790938700513335557070325628, 3.29298652740753695701377317054, 4.30032808357235128666275695206, 5.35708181227167532787761077019, 6.51914956387136270852546336776, 6.67717098920958314152394624363, 8.044107996050708391409834608218, 8.667891024523460275253883213989, 9.081650781769527718108637078862
