| L(s) = 1 | − 5.86e14·2-s − 2.90e29·4-s − 2.74e34·5-s + 3.47e41·7-s + 5.41e44·8-s + 1.60e49·10-s + 4.95e51·11-s + 1.21e55·13-s − 2.03e56·14-s − 1.33e59·16-s + 3.97e60·17-s − 4.47e62·19-s + 7.97e63·20-s − 2.90e66·22-s + 4.53e67·23-s − 8.23e68·25-s − 7.11e69·26-s − 1.00e71·28-s + 5.62e71·29-s + 7.11e73·31-s − 2.65e74·32-s − 2.33e75·34-s − 9.53e75·35-s + 3.44e77·37-s + 2.62e77·38-s − 1.48e79·40-s − 2.52e78·41-s + ⋯ |
| L(s) = 1 | − 0.736·2-s − 0.458·4-s − 0.691·5-s + 0.510·7-s + 1.07·8-s + 0.508·10-s + 1.40·11-s + 0.879·13-s − 0.375·14-s − 0.331·16-s + 0.492·17-s − 0.225·19-s + 0.316·20-s − 1.03·22-s + 1.78·23-s − 0.521·25-s − 0.647·26-s − 0.233·28-s + 0.229·29-s + 1.07·31-s − 0.829·32-s − 0.362·34-s − 0.353·35-s + 0.815·37-s + 0.165·38-s − 0.742·40-s − 0.0370·41-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
| \(L(50)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{101}{2})\) |
|
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 | \( 1 + 5.86e14T + 6.33e29T^{2} \) |
| 5 | \( 1 + 2.74e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 3.47e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 4.95e51T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.21e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 3.97e60T + 6.52e121T^{2} \) |
| 19 | \( 1 + 4.47e62T + 3.95e126T^{2} \) |
| 23 | \( 1 - 4.53e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 5.62e71T + 5.98e144T^{2} \) |
| 31 | \( 1 - 7.11e73T + 4.41e147T^{2} \) |
| 37 | \( 1 - 3.44e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 2.52e78T + 4.63e159T^{2} \) |
| 43 | \( 1 - 8.67e80T + 5.16e161T^{2} \) |
| 47 | \( 1 + 3.44e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 2.95e85T + 5.05e170T^{2} \) |
| 59 | \( 1 + 7.60e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 8.96e87T + 5.59e176T^{2} \) |
| 67 | \( 1 - 2.62e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 6.35e91T + 1.88e183T^{2} \) |
| 73 | \( 1 + 3.15e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 1.44e94T + 7.32e187T^{2} \) |
| 83 | \( 1 + 6.97e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 1.17e96T + 9.76e192T^{2} \) |
| 97 | \( 1 + 3.50e98T + 4.90e196T^{2} \) |
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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
−8.849099566271028227502258522763, −8.142185624171968101840822199216, −7.22470621810793967700069773001, −6.08510426141437445825671841589, −4.69722256333182290570037533941, −4.09809223309320599390772629833, −3.08111652336916369118224456970, −1.37922440073443680676522397539, −1.12459025172292091210994722580, 0,
1.12459025172292091210994722580, 1.37922440073443680676522397539, 3.08111652336916369118224456970, 4.09809223309320599390772629833, 4.69722256333182290570037533941, 6.08510426141437445825671841589, 7.22470621810793967700069773001, 8.142185624171968101840822199216, 8.849099566271028227502258522763
