| L(s) = 1 | − 1.35·2-s − 3-s − 0.169·4-s − 1.31·5-s + 1.35·6-s + 2.93·8-s + 9-s + 1.78·10-s − 4.13·11-s + 0.169·12-s + 1.21·13-s + 1.31·15-s − 3.63·16-s − 7.63·17-s − 1.35·18-s − 6.01·19-s + 0.223·20-s + 5.59·22-s + 3.53·23-s − 2.93·24-s − 3.25·25-s − 1.64·26-s − 27-s − 3.42·29-s − 1.78·30-s + 1.20·31-s − 0.953·32-s + ⋯ |
| L(s) = 1 | − 0.956·2-s − 0.577·3-s − 0.0845·4-s − 0.589·5-s + 0.552·6-s + 1.03·8-s + 0.333·9-s + 0.564·10-s − 1.24·11-s + 0.0488·12-s + 0.336·13-s + 0.340·15-s − 0.908·16-s − 1.85·17-s − 0.318·18-s − 1.37·19-s + 0.0498·20-s + 1.19·22-s + 0.736·23-s − 0.599·24-s − 0.651·25-s − 0.321·26-s − 0.192·27-s − 0.636·29-s − 0.325·30-s + 0.217·31-s − 0.168·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05710571795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05710571795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 43 | \( 1 + 7.13T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 - 3.28T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6.82T + 97T^{2} \) |
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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
−8.318914237185902071188614898262, −7.45638164397375918103867397911, −6.86278655650794461994491604061, −6.09106531320975153296277020385, −5.06011448636078707058475549806, −4.53432453636295185139732085099, −3.82912168522317938107904624549, −2.51819846827693960544078551454, −1.61273240860405541332901706840, −0.14864401419505364942586242449,
0.14864401419505364942586242449, 1.61273240860405541332901706840, 2.51819846827693960544078551454, 3.82912168522317938107904624549, 4.53432453636295185139732085099, 5.06011448636078707058475549806, 6.09106531320975153296277020385, 6.86278655650794461994491604061, 7.45638164397375918103867397911, 8.318914237185902071188614898262
