| L(s) = 1 | + 3.22·5-s + 6.57i·7-s − 13.8i·11-s − 19.0·13-s − 23.0·17-s − 18.8i·19-s + 13.5i·23-s − 14.5·25-s − 0.752·29-s − 46.4i·31-s + 21.1i·35-s − 2.68·37-s + 34.1·41-s − 20.9i·43-s − 15.4i·47-s + ⋯ |
| L(s) = 1 | + 0.645·5-s + 0.938i·7-s − 1.26i·11-s − 1.46·13-s − 1.35·17-s − 0.991i·19-s + 0.591i·23-s − 0.583·25-s − 0.0259·29-s − 1.49i·31-s + 0.605i·35-s − 0.0724·37-s + 0.832·41-s − 0.486i·43-s − 0.327i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6710884168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6710884168\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.22T + 25T^{2} \) |
| 7 | \( 1 - 6.57iT - 49T^{2} \) |
| 11 | \( 1 + 13.8iT - 121T^{2} \) |
| 13 | \( 1 + 19.0T + 169T^{2} \) |
| 17 | \( 1 + 23.0T + 289T^{2} \) |
| 19 | \( 1 + 18.8iT - 361T^{2} \) |
| 23 | \( 1 - 13.5iT - 529T^{2} \) |
| 29 | \( 1 + 0.752T + 841T^{2} \) |
| 31 | \( 1 + 46.4iT - 961T^{2} \) |
| 37 | \( 1 + 2.68T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 15.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 40.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 169.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 93.1T + 9.40e3T^{2} \) |
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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
−9.357554391247993592443157159496, −9.101685831847850062816567445181, −8.032379553295088826391016300275, −7.01120055445478808193079714939, −6.00028001347191085524179937952, −5.40939147421134349912422229756, −4.31792525216372395775667378277, −2.80695185863751909509632940979, −2.12429536201764548637511788926, −0.20069266792465771188868627119,
1.63357783994312492565393619971, 2.62158357252990220999774832749, 4.21241558659802511578647993113, 4.77452046208231275807539345199, 5.99011348718323664612090291601, 7.05998846382149993196432025560, 7.44405753663031079766884768756, 8.689574652394480420881146395997, 9.689580303527192600181648016443, 10.14290457645645746636580332215
