| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 4·11-s + 4·13-s + 4·14-s + 16-s + 20-s + 4·22-s + 4·23-s + 25-s − 4·26-s − 4·28-s − 6·29-s − 4·31-s − 32-s − 4·35-s + 2·37-s − 40-s − 2·41-s − 4·44-s − 4·46-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.158·40-s − 0.312·41-s − 0.603·44-s − 0.589·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.479738235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479738235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.06521915960638, −12.83601699494820, −12.53943986787198, −11.63866158776761, −11.25375979104225, −10.69170444587400, −10.32673073123349, −9.976205708260562, −9.254456061671669, −9.119782863937126, −8.580299761652649, −7.913781159908712, −7.458337697793174, −6.886599964280091, −6.503081550314160, −5.895967164409062, −5.544086912110370, −5.018409164886578, −4.040005357637791, −3.463948057052487, −3.143737826839549, −2.330667977120323, −2.013956027437776, −0.9150288490424362, −0.4865414357970641,
0.4865414357970641, 0.9150288490424362, 2.013956027437776, 2.330667977120323, 3.143737826839549, 3.463948057052487, 4.040005357637791, 5.018409164886578, 5.544086912110370, 5.895967164409062, 6.503081550314160, 6.886599964280091, 7.458337697793174, 7.913781159908712, 8.580299761652649, 9.119782863937126, 9.254456061671669, 9.976205708260562, 10.32673073123349, 10.69170444587400, 11.25375979104225, 11.63866158776761, 12.53943986787198, 12.83601699494820, 13.06521915960638
