| L(s) = 1 | + 0.801·2-s − 3·3-s − 7.35·4-s + 20.8·5-s − 2.40·6-s + 14.6·7-s − 12.3·8-s + 9·9-s + 16.7·10-s + 69.6·11-s + 22.0·12-s − 4.18·13-s + 11.7·14-s − 62.5·15-s + 48.9·16-s − 83.8·17-s + 7.21·18-s − 24.8·19-s − 153.·20-s − 43.8·21-s + 55.8·22-s − 23·23-s + 36.9·24-s + 309.·25-s − 3.35·26-s − 27·27-s − 107.·28-s + ⋯ |
| L(s) = 1 | + 0.283·2-s − 0.577·3-s − 0.919·4-s + 1.86·5-s − 0.163·6-s + 0.788·7-s − 0.544·8-s + 0.333·9-s + 0.528·10-s + 1.90·11-s + 0.530·12-s − 0.0893·13-s + 0.223·14-s − 1.07·15-s + 0.765·16-s − 1.19·17-s + 0.0945·18-s − 0.299·19-s − 1.71·20-s − 0.455·21-s + 0.541·22-s − 0.208·23-s + 0.314·24-s + 2.47·25-s − 0.0253·26-s − 0.192·27-s − 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.658157518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658157518\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 0.801T + 8T^{2} \) |
| 5 | \( 1 - 20.8T + 125T^{2} \) |
| 7 | \( 1 - 14.6T + 343T^{2} \) |
| 11 | \( 1 - 69.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.18T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 252.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 21.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 4.65T + 1.48e5T^{2} \) |
| 59 | \( 1 + 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 61.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 615.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 295.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 48.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 470.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 50.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 9.17T + 7.04e5T^{2} \) |
| 97 | \( 1 + 431.T + 9.12e5T^{2} \) |
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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
−14.08191050809862734386179832193, −13.36906398136735245311096382556, −12.19426772070960206684666331849, −10.84051515298872681501679773032, −9.518124798370091115602919175928, −8.892910743681534021859256320510, −6.58190194537437301865220843740, −5.58902167351863905511184642244, −4.35747050757119671683645719706, −1.61030033591133727410187809762,
1.61030033591133727410187809762, 4.35747050757119671683645719706, 5.58902167351863905511184642244, 6.58190194537437301865220843740, 8.892910743681534021859256320510, 9.518124798370091115602919175928, 10.84051515298872681501679773032, 12.19426772070960206684666331849, 13.36906398136735245311096382556, 14.08191050809862734386179832193
