| L(s) = 1 | + 270.·2-s − 3.48e3·3-s − 4.51e5·4-s + 4.93e6·5-s − 9.40e5·6-s + 4.03e7·7-s − 2.63e8·8-s − 1.15e9·9-s + 1.33e9·10-s + 1.74e9·11-s + 1.57e9·12-s − 5.98e10·13-s + 1.09e10·14-s − 1.71e10·15-s + 1.65e11·16-s − 3.16e11·17-s − 3.10e11·18-s − 1.99e12·19-s − 2.22e12·20-s − 1.40e11·21-s + 4.72e11·22-s − 1.27e13·23-s + 9.17e11·24-s + 5.23e12·25-s − 1.61e13·26-s + 8.04e12·27-s − 1.82e13·28-s + ⋯ |
| L(s) = 1 | + 0.373·2-s − 0.102·3-s − 0.860·4-s + 1.12·5-s − 0.0380·6-s + 0.377·7-s − 0.694·8-s − 0.989·9-s + 0.421·10-s + 0.223·11-s + 0.0878·12-s − 1.56·13-s + 0.141·14-s − 0.115·15-s + 0.601·16-s − 0.648·17-s − 0.369·18-s − 1.41·19-s − 0.971·20-s − 0.0385·21-s + 0.0833·22-s − 1.47·23-s + 0.0708·24-s + 0.274·25-s − 0.583·26-s + 0.203·27-s − 0.325·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 4.03e7T \) |
| good | 2 | \( 1 - 270.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 3.48e3T + 1.16e9T^{2} \) |
| 5 | \( 1 - 4.93e6T + 1.90e13T^{2} \) |
| 11 | \( 1 - 1.74e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 5.98e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 3.16e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.99e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.27e13T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.54e14T + 6.10e27T^{2} \) |
| 31 | \( 1 + 3.26e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 3.42e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.02e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 4.43e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 6.50e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.73e15T + 5.77e32T^{2} \) |
| 59 | \( 1 - 4.67e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.00e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 4.36e16T + 4.95e34T^{2} \) |
| 71 | \( 1 + 6.23e16T + 1.49e35T^{2} \) |
| 73 | \( 1 + 7.23e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 3.58e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 7.37e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 3.96e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 7.91e18T + 5.60e37T^{2} \) |
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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
−17.20393125982648621099576074079, −14.64803826581116402709672041392, −13.79562366661418900502232698420, −12.22477876007795019680136049185, −10.06930587477902549280815100670, −8.620806180965386741682291782578, −6.03884131910157386436984850110, −4.65840038934072170891488764601, −2.34690008206409307427150395869, 0,
2.34690008206409307427150395869, 4.65840038934072170891488764601, 6.03884131910157386436984850110, 8.620806180965386741682291782578, 10.06930587477902549280815100670, 12.22477876007795019680136049185, 13.79562366661418900502232698420, 14.64803826581116402709672041392, 17.20393125982648621099576074079
