| L(s) = 1 | − 2.49·2-s + 1.67·3-s − 1.79·4-s − 22.2·5-s − 4.17·6-s + 24.3·8-s − 24.1·9-s + 55.4·10-s − 31.8·11-s − 3.00·12-s + 42.6·13-s − 37.2·15-s − 46.4·16-s − 43.3·17-s + 60.2·18-s − 27.6·19-s + 39.8·20-s + 79.4·22-s − 129.·23-s + 40.9·24-s + 369.·25-s − 106.·26-s − 85.8·27-s − 13.0·29-s + 92.9·30-s − 206.·31-s − 79.4·32-s + ⋯ |
| L(s) = 1 | − 0.880·2-s + 0.322·3-s − 0.223·4-s − 1.98·5-s − 0.284·6-s + 1.07·8-s − 0.895·9-s + 1.75·10-s − 0.874·11-s − 0.0722·12-s + 0.909·13-s − 0.641·15-s − 0.725·16-s − 0.618·17-s + 0.789·18-s − 0.334·19-s + 0.445·20-s + 0.770·22-s − 1.17·23-s + 0.347·24-s + 2.95·25-s − 0.801·26-s − 0.611·27-s − 0.0834·29-s + 0.565·30-s − 1.19·31-s − 0.438·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 + 2.49T + 8T^{2} \) |
| 3 | \( 1 - 1.67T + 27T^{2} \) |
| 5 | \( 1 + 22.2T + 125T^{2} \) |
| 11 | \( 1 + 31.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 13.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 328.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 469.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 303.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 549.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 878.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 380.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 440.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 911.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 416.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
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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
−8.340106699083008632124134447118, −7.88109561662020273459041126257, −7.31408238688927126751665856347, −6.11892566252150041664405611441, −4.94111938302073205531516842580, −4.09275360138318871741095550101, −3.50741153104448800347723854340, −2.31243389515471160137078891895, −0.71645100334853856938345374463, 0,
0.71645100334853856938345374463, 2.31243389515471160137078891895, 3.50741153104448800347723854340, 4.09275360138318871741095550101, 4.94111938302073205531516842580, 6.11892566252150041664405611441, 7.31408238688927126751665856347, 7.88109561662020273459041126257, 8.340106699083008632124134447118
