| L(s) = 1 | + 7.83e13·2-s − 6.27e29·4-s − 7.38e34·5-s + 1.10e42·7-s − 9.88e43·8-s − 5.78e48·10-s + 1.49e51·11-s + 1.26e55·13-s + 8.65e55·14-s + 3.90e59·16-s − 1.42e61·17-s + 6.64e62·19-s + 4.63e64·20-s + 1.16e65·22-s − 1.16e67·23-s + 3.86e69·25-s + 9.88e68·26-s − 6.93e71·28-s + 3.35e72·29-s + 1.00e74·31-s + 9.31e73·32-s − 1.11e75·34-s − 8.15e76·35-s + 5.27e77·37-s + 5.20e76·38-s + 7.29e78·40-s + 9.79e79·41-s + ⋯ |
| L(s) = 1 | + 0.0983·2-s − 0.990·4-s − 1.85·5-s + 1.62·7-s − 0.195·8-s − 0.182·10-s + 0.421·11-s + 0.913·13-s + 0.159·14-s + 0.971·16-s − 1.76·17-s + 0.334·19-s + 1.84·20-s + 0.0414·22-s − 0.456·23-s + 2.45·25-s + 0.0898·26-s − 1.60·28-s + 1.37·29-s + 1.51·31-s + 0.291·32-s − 0.173·34-s − 3.02·35-s + 1.24·37-s + 0.0328·38-s + 0.363·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(100-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+99/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(50)\) |
\(\approx\) |
\(2.048358264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.048358264\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 7.83e13T + 6.33e29T^{2} \) |
| 5 | \( 1 + 7.38e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 1.10e42T + 4.62e83T^{2} \) |
| 11 | \( 1 - 1.49e51T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.26e55T + 1.90e110T^{2} \) |
| 17 | \( 1 + 1.42e61T + 6.52e121T^{2} \) |
| 19 | \( 1 - 6.64e62T + 3.95e126T^{2} \) |
| 23 | \( 1 + 1.16e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 3.35e72T + 5.98e144T^{2} \) |
| 31 | \( 1 - 1.00e74T + 4.41e147T^{2} \) |
| 37 | \( 1 - 5.27e77T + 1.78e155T^{2} \) |
| 41 | \( 1 - 9.79e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 3.61e80T + 5.16e161T^{2} \) |
| 47 | \( 1 + 4.58e81T + 3.44e165T^{2} \) |
| 53 | \( 1 - 1.68e85T + 5.05e170T^{2} \) |
| 59 | \( 1 - 2.66e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 1.56e88T + 5.59e176T^{2} \) |
| 67 | \( 1 - 1.29e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 5.47e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 2.27e92T + 2.94e184T^{2} \) |
| 79 | \( 1 - 1.20e94T + 7.32e187T^{2} \) |
| 83 | \( 1 - 1.20e95T + 9.74e189T^{2} \) |
| 89 | \( 1 + 1.14e96T + 9.76e192T^{2} \) |
| 97 | \( 1 - 1.82e97T + 4.90e196T^{2} \) |
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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
−9.018454638224502158526452502461, −8.260386130953213858907401299382, −7.88586007431606192139038372707, −6.52679251823800707998487137275, −5.01804692064955576237126834945, −4.20201871330984116887619723788, −4.07515013667035854660276966679, −2.61473841901202756420355386807, −1.07066613890197949229522356721, −0.63513328756066111344813396074,
0.63513328756066111344813396074, 1.07066613890197949229522356721, 2.61473841901202756420355386807, 4.07515013667035854660276966679, 4.20201871330984116887619723788, 5.01804692064955576237126834945, 6.52679251823800707998487137275, 7.88586007431606192139038372707, 8.260386130953213858907401299382, 9.018454638224502158526452502461
