| L(s) = 1 | + 4.84·2-s − 3·3-s + 15.5·4-s + 2.99·5-s − 14.5·6-s + 20.4·7-s + 36.4·8-s + 9·9-s + 14.5·10-s − 36.3·11-s − 46.5·12-s − 34.9·13-s + 98.9·14-s − 8.97·15-s + 52.4·16-s − 37.0·17-s + 43.6·18-s − 36.1·19-s + 46.4·20-s − 61.2·21-s − 176.·22-s − 23·23-s − 109.·24-s − 116.·25-s − 169.·26-s − 27·27-s + 316.·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.577·3-s + 1.93·4-s + 0.267·5-s − 0.989·6-s + 1.10·7-s + 1.60·8-s + 0.333·9-s + 0.458·10-s − 0.995·11-s − 1.11·12-s − 0.746·13-s + 1.88·14-s − 0.154·15-s + 0.820·16-s − 0.528·17-s + 0.571·18-s − 0.436·19-s + 0.518·20-s − 0.636·21-s − 1.70·22-s − 0.208·23-s − 0.929·24-s − 0.928·25-s − 1.27·26-s − 0.192·27-s + 2.13·28-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
| \(L(2)\) |
\(\approx\) |
\(3.168755616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.168755616\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.84T + 8T^{2} \) |
| 5 | \( 1 - 2.99T + 125T^{2} \) |
| 7 | \( 1 - 20.4T + 343T^{2} \) |
| 11 | \( 1 + 36.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 512.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 519.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 336.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 117.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 60.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 754.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 180.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 378.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 992.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 358.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 808.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 583.T + 9.12e5T^{2} \) |
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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
−14.09722205919435692905815909387, −13.23083924032645697045985557571, −12.15892956079544120781003556274, −11.34896013425153674700784036109, −10.24475840585504020114467712668, −7.973713374012432364804948799022, −6.51342734240527288090478083937, −5.25882235255450273221902591486, −4.47502204378144636035548924478, −2.36712814468245378589920083027,
2.36712814468245378589920083027, 4.47502204378144636035548924478, 5.25882235255450273221902591486, 6.51342734240527288090478083937, 7.973713374012432364804948799022, 10.24475840585504020114467712668, 11.34896013425153674700784036109, 12.15892956079544120781003556274, 13.23083924032645697045985557571, 14.09722205919435692905815909387
