| L(s) = 1 | + 48.4·2-s − 518.·3-s − 5.84e3·4-s − 2.56e4·5-s − 2.51e4·6-s + 1.36e5·7-s − 6.80e5·8-s − 1.32e6·9-s − 1.24e6·10-s + 3.19e6·11-s + 3.02e6·12-s + 9.81e6·13-s + 6.61e6·14-s + 1.32e7·15-s + 1.49e7·16-s − 3.42e7·17-s − 6.42e7·18-s + 3.43e8·19-s + 1.49e8·20-s − 7.07e7·21-s + 1.54e8·22-s + 1.48e8·23-s + 3.52e8·24-s − 5.63e8·25-s + 4.75e8·26-s + 1.51e9·27-s − 7.97e8·28-s + ⋯ |
| L(s) = 1 | + 0.535·2-s − 0.410·3-s − 0.713·4-s − 0.733·5-s − 0.219·6-s + 0.438·7-s − 0.917·8-s − 0.831·9-s − 0.392·10-s + 0.543·11-s + 0.292·12-s + 0.564·13-s + 0.234·14-s + 0.301·15-s + 0.222·16-s − 0.343·17-s − 0.445·18-s + 1.67·19-s + 0.523·20-s − 0.180·21-s + 0.290·22-s + 0.208·23-s + 0.376·24-s − 0.461·25-s + 0.302·26-s + 0.751·27-s − 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.367752502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367752502\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 1.48e8T \) |
| good | 2 | \( 1 - 48.4T + 8.19e3T^{2} \) |
| 3 | \( 1 + 518.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 2.56e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 1.36e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 3.19e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 9.81e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 3.42e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.43e8T + 4.20e16T^{2} \) |
| 29 | \( 1 + 8.98e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 4.72e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.52e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 7.18e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 7.07e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 4.46e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 7.70e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.01e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 3.20e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.74e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 3.97e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.92e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.81e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.09e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 3.83e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 3.83e12T + 6.73e25T^{2} \) |
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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
−14.57502796520060021163811430076, −13.61799815249017293618810838495, −12.06607599614458091778791240205, −11.27423302409575505821435158056, −9.283134967630457003675299609168, −7.977668666476023616452024374045, −5.98549750852219987393004993513, −4.67096976819465186111724540483, −3.32268838894649674028101917487, −0.73659012959601582120775053071,
0.73659012959601582120775053071, 3.32268838894649674028101917487, 4.67096976819465186111724540483, 5.98549750852219987393004993513, 7.977668666476023616452024374045, 9.283134967630457003675299609168, 11.27423302409575505821435158056, 12.06607599614458091778791240205, 13.61799815249017293618810838495, 14.57502796520060021163811430076
