| L(s) = 1 | + 2.09e6·2-s − 3.29e10·3-s + 4.39e12·4-s + 3.97e14·5-s − 6.91e16·6-s + 1.80e18·7-s + 9.22e18·8-s + 7.58e20·9-s + 8.32e20·10-s − 3.33e22·11-s − 1.44e23·12-s − 7.96e23·13-s + 3.78e24·14-s − 1.30e25·15-s + 1.93e25·16-s − 3.35e26·17-s + 1.58e27·18-s + 6.16e26·19-s + 1.74e27·20-s − 5.94e28·21-s − 6.98e28·22-s + 2.43e29·23-s − 3.04e29·24-s − 9.79e29·25-s − 1.67e30·26-s − 1.41e31·27-s + 7.92e30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.372·5-s − 1.28·6-s + 1.21·7-s + 0.353·8-s + 2.30·9-s + 0.263·10-s − 1.35·11-s − 0.909·12-s − 0.894·13-s + 0.862·14-s − 0.677·15-s + 0.250·16-s − 1.17·17-s + 1.63·18-s + 0.198·19-s + 0.186·20-s − 2.21·21-s − 0.959·22-s + 1.28·23-s − 0.643·24-s − 0.861·25-s − 0.632·26-s − 2.38·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.09e6T \) |
| good | 3 | \( 1 + 3.29e10T + 3.28e20T^{2} \) |
| 5 | \( 1 - 3.97e14T + 1.13e30T^{2} \) |
| 7 | \( 1 - 1.80e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 3.33e22T + 6.02e44T^{2} \) |
| 13 | \( 1 + 7.96e23T + 7.93e47T^{2} \) |
| 17 | \( 1 + 3.35e26T + 8.11e52T^{2} \) |
| 19 | \( 1 - 6.16e26T + 9.69e54T^{2} \) |
| 23 | \( 1 - 2.43e29T + 3.58e58T^{2} \) |
| 29 | \( 1 + 2.03e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 1.01e32T + 1.34e64T^{2} \) |
| 37 | \( 1 + 4.91e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 5.16e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 9.91e34T + 1.73e70T^{2} \) |
| 47 | \( 1 - 2.80e35T + 7.94e71T^{2} \) |
| 53 | \( 1 - 6.66e36T + 1.39e74T^{2} \) |
| 59 | \( 1 - 4.38e37T + 1.40e76T^{2} \) |
| 61 | \( 1 + 8.24e37T + 5.87e76T^{2} \) |
| 67 | \( 1 + 2.28e39T + 3.32e78T^{2} \) |
| 71 | \( 1 - 6.57e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 1.46e40T + 1.32e80T^{2} \) |
| 79 | \( 1 - 4.38e40T + 3.96e81T^{2} \) |
| 83 | \( 1 + 2.02e41T + 3.31e82T^{2} \) |
| 89 | \( 1 + 3.53e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 6.40e42T + 2.69e85T^{2} \) |
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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
−17.05310831717640847041831092823, −15.35850698157405772267829422091, −13.10214823697675250405292929622, −11.59140817763000318065709372324, −10.55575780948299638664076831589, −7.20266994926007741624699724485, −5.45269944989909920854280068663, −4.78468723426506706567837544926, −1.85459738274636160704058680932, 0,
1.85459738274636160704058680932, 4.78468723426506706567837544926, 5.45269944989909920854280068663, 7.20266994926007741624699724485, 10.55575780948299638664076831589, 11.59140817763000318065709372324, 13.10214823697675250405292929622, 15.35850698157405772267829422091, 17.05310831717640847041831092823
